diff --git "a/9NFLT4oBgHgl3EQfty_-/content/tmp_files/load_file.txt" "b/9NFLT4oBgHgl3EQfty_-/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/9NFLT4oBgHgl3EQfty_-/content/tmp_files/load_file.txt" @@ -0,0 +1,2752 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf,len=2751 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12153v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='AP] 28 Jan 2023 WELL-POSEDNESS OF THE 3D PESKIN PROBLEM EDUARDO GARC´IA-JU´AREZ˚, PO-CHUN KUO:,;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', YOICHIRO MORI:,§, AND ROBERT M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN:,¶ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This paper introduces the 3D Peskin problem: a two-dimensional elastic membrane immersed in a three-dimensional steady Stokes flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We obtain the equations that model this free boundary problem and show that they admit a boundary integral reduction, providing an evolution equation for the elastic interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We consider general nonlinear elastic laws, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', the fully nonlinear Peskin problem, and prove that the problem is well-posed in low-regularity H¨older spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, we prove that the elastic membrane becomes smooth instantly in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Introduction 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Formulation and Boundary Integral Reduction 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Preliminaries 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Nonlinear decomposition 18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Calculus estimates 24 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Frozen-coefficient Semigroup 39 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Local well-posedness 56 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Higher Regularity 66 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Besov Spaces and Fourier Multiplier Theorems 75 Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Estimates for the semigroup e´tLApξq 77 References 90 ˚Departamento de An´alisis Matem´atico, Universidad de Sevilla, C/Tarfia s/n, Cam- pus Reina Mercedes, 41012, Sevilla, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' egarciajuarez@ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='edu :Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', 209 South 33rd St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', Philadelphia, PA 19104, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='kuopo@sas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='upenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='edu §y1mori@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='upenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='edu ¶strain@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='upenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='edu Date: January 31, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 35Q35, 35C15, 35R11, 35R35, 76D07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Peskin problem, 3D, Fluid-Structure Interaction, immersed boundary problem, Stokes flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ˚supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sk�lodowska-Curie grant agreement CAMINFLOW No 101031111, and the AEI project PID2021-125021NAI00 (Spain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='partially supported by NSF grant DMS-2042144 (USA) awarded to YM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' §partially supported by the NSF grant DMS-1907583, 2042144 (USA) and the Math+X award from the Simons Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ¶partially supported by the NSF grants DMS-1764177 and DMS-2055271 (USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Introduction The immersed boundary method, introduced by Peskin [40, 41] to study the blood flow around heart valves, has been widely applied to numerically study fluid- structure interaction (FSI) problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' These FSI problems, in which a fluid interacts with elastic structures, appear naturally in many engineering and biophysics appli- cations [44,46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Despite their importance, both the computational methods and the FSI problems themselves are poorly understood from an analytical standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A major impediment has been the lack of analytical understanding of the underlying PDEs, which are typically nonlinear and nonlocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Results are particularly scarce in the more realistic three-dimensional settings, where the coupling of nonlocal effects with non-trivial geometry substantially increases the complexity of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since the recent breakthrough works [34] and [37], which provided the strong solution theory for the problem of an immersed elastic string in a two-dimensional fluid, the so-called 2D Peskin problem has attracted a lot of attention [8,9,23,25, 33,51,52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In this paper, we initiate the study of its three-dimensional counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We introduce the formulation and develop the well-posedness theory for the three- dimensional (fully nonlinear) Peskin problem of an elastic membrane immersed in a fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Description of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We consider the following problem in which a three-dimensional incompressible Stokes fluid interacts with an elastic membrane in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A closed elastic interface Γ encloses a simply connected bounded domain Ω Ă R3 filled with a Stokes fluid with viscosity µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The outside region R3zΩ is filled with a Stokes fluid of viscosity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The equations satisfied are: µ∆u ´ ∇p “ 0 in Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) ∆u ´ ∇p “ 0 in R3zΩ, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) ∇ ¨ u “ 0 in R3zΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) Here u is the velocity field and p is the pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We impose the following condition in the far field: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) u Ñ 0 as |x| Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We supplement the above with interface conditions on the time-evolving surface Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For any quantity w defined on Ω and R3zΩ, we set: �w� “ w|Γi ´ w|Γe where w|Γi,e are the trace values of w at Γ evaluated from the Ω (interior) and R3zΩ (exterior) sides of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let n be the outward pointing unit normal vector on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The interface conditions are: �u� “ 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) �Σn� “ F el, Σ “ # µ ` ∇u ` p∇uqT˘ ´ pI in Ω ∇u ` p∇uqT ´ pI in R3zΩ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) BX Bt “ upX, tq, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) where I is the 3 ˆ 3 identity matrix and X : S2 ÞÑ Γptq the map that describes the evolving membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This map gives the deformation of the reference configuration S2, the standard embedding of the sphere of radius 1 in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The first condition WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 3 is the no-slip boundary condition and the second is the stress balance condition where Σ is the fluid stress and F el is the elastic force exerted by the interface Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The last condition states that the membrane evolves with the fluid flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Note that the elastic surface Γ “ Γptq and hence Ω “ Ωptq changes with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Once given the constitutive equation for the elastic force F el, equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) form the so-called jump formulation of the 3D Peskin problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let pg and g denote the metric tensors on S2 and Γ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A natural choice for the elastic stretching force is given by [19,28] (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) F el “ a detppg´1gq∇S2 ¨ T p∇S2Xq, where T p∇S2Xq :“ T p|∇S2X|q |∇S2X| ∇S2X “: T p|∇S2X|q∇S2X, ∇S2 denotes the surface gradient on S2, |A| denotes the Frobenius norm of matrix A, and T has to satisfy T ą 0, dT {dλ ě 0 (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 for further notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In Section 2 more details about the derivation of the elastic force are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For a Hookean material, T is linear and hence the elastic force is linear in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will consider general T , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', the fully nonlinear Peskin problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Compared to fluid interface problems, such as a drop of liquid surrounded by another fluid or vacuum [17,43,45,49,50], where only the shape of the interface mat- ters, here it is not expected that Eulerian methods on their own should suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Due to the elastic nature of the membrane, the stretching, given by the parametriza- tion, has a strong influence on the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, one needs to keep track of the membrane configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lagrangian methods are needed, making it harder to work in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In particular, one cannot freely reparametrize the surface, an idea frequently used to obtain extra cancellations in the study of fluid interfaces [13,22,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' An important feature of the Peskin problem is that it admits a Boundary Integral formulation, whose derivation is given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' When µ “ 1, the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1)- (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) is equivalent to the following evolution equation for X: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) BX Bt ppxq “ ż S2GpXppxq ´ Xppyqq∇S2 ¨ ´ T p|∇S2Xppyq|q ∇S2Xppyq |∇S2Xppyq| ¯ dµS2ppyq, Xppxq|t“0 “ X0ppxq, where Gpxq is the Stokeslet tensor in R3: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) Gpxq “ 1 8π ´ 1 |x|I3 ` x b x |x|3 ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We have suppressed the dependence of X on t to avoid cluttered notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence- forth, we will assume µ “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It will be sometimes convenient in the analysis to work with coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let θ “ pθ1, θ2q be a (local) coordinate system on S2 and let px “ x Xpθq P S2 Ă R3 be the point on S2 corresponding to θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let Xpθq “ Xpx Xpθqq P Γ Ă R3 be the position on Γ corresponding to the coordinate point θ (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If px “ x Xpθq, we will write Xppxq and Xpθq in an abuse of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, after integration by parts and choosing an isothermal coordinate system, equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) becomes BX Bt pθq “ ´p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ż R2 B Bηi GpXpθq´Xpηqq ˜F el,ipXqpηqdη1dη2, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN where we denote ˜F el,ipXqpηq “ T pλpηqq λpηq B Bηi Xpηq, λpηq “ a trppg´1pηqgpηqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Above we use the explicit definitions of pg and g given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Deformation map Xp¨, tq : S2 Ñ Γptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Some important properties of the solutions to the Peskin problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) are easier to deduce from the jump formulation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The incompressibility condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3), together with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7), implies the conservation of the volume of the enclosed region Ω: d dt|Ωptq| “ 0, |Ωptq| “ 1 3 ż S2 Xppxq ¨ nppxqdµΓptqppxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, the elastic energy defined as follows EpXq “ ż S2 AEp|∇S2Xppxq|qdµS2ppxq, A1 Epλq “ T pλq, satisfies the balance d dtEpXq “ ´ ż R3 |∇u|2dx, which shows that the elastic energy is dissipated due to the viscosity of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This relation follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7), integration by parts, and using conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2), and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5), consecutively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For a linear elasticity law, the elastic energy is the 9H1pS2q norm of the interface, EpXq “ 1 2 ż S2 |∇S2Xppxq|2dµS2ppxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A third important property of the Peskin problem is that it satisfies a scaling invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We must first mention that the definition of solutions requires that the WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 5 interface is non-degenerate and does not self-intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This is typically enforced through the arc-chord condition: |X|˚ :“ inf px‰py px, pyPS2 |X ppxq ´ X ppyq| |px ´ py| ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If Xppx, tq solves (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9), then, for any λ ą 0, Xλppx, tq :“ λ´1Xpλpx, λtq also solves the equation, and |Xλ|˚ “ |X|˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, 9C1pS2q and spaces with the same scaling, such as 9H2pS2q, are critical spaces for 3D Peskin problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Notice that the energy balance above only gives control of the 9H1pS2q norm, hence the Peskin problem is supercritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The formulation of the problem, both in jump and Boundary Integral forms, is derived in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Once the formulation is provided, the main objective of the paper is to show that the problem is well-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' More specifically, we will first show the existence and uniqueness of strong solutions with initial data in little H¨older spaces, h1,γpS2q, γ P p0, 1q, defined as the completion of the set of smooth functions in C1,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 (Strong solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let X P Cpr0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C1,γpS2qqXC1pr0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' CγpS2qq, γ P p0, 1q, and |Xptq|˚ ą 0 for t P r0, T s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, X is a strong solution to the 3D Peskin problem with initial data Xp0q “ X0 if it satisfies equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) for t P p0, T s and Xptq Ñ X0 in C1,γpS2q as t Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The choice of little H¨older spaces will be needed to obtain the convergence to the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In Section 7 we will prove the following: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Consider the 3D Peskin problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) with initial data satisfying X0 P h1,γpS2q, |X0|˚ ą 0, and T P C3 such that T ą 0, dT {dλ ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, there exists some time T ą 0 such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) has a unique strong solution X, X P Cpr0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' h1,γpS2qq X C1pr0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' hγpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It is instructive to briefly recall the idea of the proof for the 2D linear Peskin problem [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For X a non-degenerate, closed simple plane curve, the boundary integral formulation in 2D is given by BtXpθ, tq “ ´ ż S1 BηGpXpθq ´ XpηqqBηXpηqdη, where G is the Stokeslet in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It turns out that one can perform a small-scale decomposition [30,37] to write it as follows BtX “ 1 4ΛX ` RpXq, ΛX “ HBθX, with RpXq a lower order operator compared to Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, it is natural to construct the solution as a fixed point of the equation written in Duhamel form: Xptq “ etΛX0 ` ż t 0 ept´τqΛRpXpτqqdτ We notice two important facts: the semigroup is explicit, both in space and Fourier variables, and the equation is semilinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Even for nonlinear elastic law, the lead- ing term has a kernel not depending on the curve itself, ´ 1 4HpT p|∇S2X|q∇S2Xq, making it possible to use the Λ-like structure via energy methods [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Here, we first consider the strategy adapted to nonlinear equations in [35] (see also [47] for 2D Peskin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let us write equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) as follows BX Bt “ FpXq, X|t“0 “ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, at least formally, linearization around the initial data would give (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) Xptq “ eLpX0qtX0 ` ż t 0 eLpX0qpt´τqEpXpτqqdτ, with LpX0q “ BXFpX0qX the Gateaux derivative of F at X0 and EpXq “ FpXq ´ LpX0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, while EpXq is not expected to be smoother than LpX0q, it should be small for short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' However, one first need to make sense of the above expression (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) by showing that LpX0q generates an analytic semigroup, which amounts to proving that the operator is sectorial in adequate spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This is the core of the abstract Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1, whose proof encompasses a fixed point argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The application of this theorem to our problem soon becomes highly involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This is done in Propositions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since the equation is not semilinear, the process will require to further decompose the operator LpX0q and then freeze the coefficients at a given point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The decomposition must be done maintaining a derivative structure for the kernel that allows extra cancellations, required to control the singular integral operators that appear, and so that we can invert the frozen-coefficient operator (the study of this part is done separately in Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Schematically, we decompose the kernel in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) as follows B Bηi ` GpXpθq´Xpηqq ˘ « ´ B Bxj G p∇Xpηqpθ ´ ηqq BXj Bηi pηq ` RpXqpθq « 1 8π BXpηq Bηi ¨ p∇Xpηqpθ´ηqq |∇Xpηqpθ´ηq|3 ` ¨ ¨ ¨ ` RpXqpθq, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) where one expects RpXq to be lower order and the dots represent additional terms of high order coming from the second term in Gppxq (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) (we note that in 2D these additional high-order terms cancel each other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' These leading kernels are not of convolution type and cannot be written as a derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For this purpose, one could be tempted to use ∇Xpθq in the approximation instead of ∇Xpηq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' However, higher derivatives of X would appear later in the proof and the argument would not close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, to take advantage of the derivative structure, we will be forced to estimate together the leading and remainder terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In a second step, we approximate ∇Xpηq in the leading kernels above by its value at a given point (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10 for more details), which requires the introduction of a partition of unity for the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Due to the geometry of the problem, we need to work with charts, and due to the nonlocal character of the equation a second localization procedure will be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A fine implementation of these localization procedures will be crucial to avoid transition maps that would otherwise overcomplicate the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the fully nonlinear Peskin problem, we must linearize and freeze the coeffi- cient of the elastic force as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2, we show that the frozen-coefficient linear operator in the general force case is given by pLAY qkpθq “ ´ ż R2 B Bηi pGk,lpA pθ ´ ηqqqpTF pAq∇Y ql,ipηqdη1dη2, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 7 where A is a constant matrix and TFpAq a tensor (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, in the general case, the multiplier for the frozen-coefficient linear operator becomes: LApξq “ I ` vpξq b vpξq 4detpBq |B´1ξ| ˆT p∥A∥F q ∥A∥F ˆ |ξ|2 I´ Aξ b Aξ ∥A∥F ˙ ` dT dλ p∥A∥F qAξ b Aξ ∥A∥F ˙ , where || ¨ ||F denotes the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It is not difficult to see that, if T ą 0 and dT {dλ ě 0, then the above is coercive in |ξ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, in contrast to the 2D case, dT {dλ “ 0 is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In fact, if T satisfies T ą 0 and pdT {dλq{T ą ´1, then the problem is expected to be locally well-posed if the initial condition is sufficiently close to the uniform sphere (pg´1g is close to a multiple of the identity matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This is an interesting difference between 2D and 3D Peskin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will use this operator (in conjunction with the localization procedures) to show that the full operator LpX0q “ BXFpX0qX is sectorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The approximation in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) is done on the equation written in coordinates partly to obtain a linear leading operator given by a Fourier multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, we notice that the regularity obtained in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2 for the strong solu- tions is not enough to satisfy equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) in a classical sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Obtaining higher regularity for the solutions is also important since this further regularity is needed for the equivalence between different formulations to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The abstract theory for nonlinear equations in [35] does not yield gain of smoothness for the solution, and in fact this important point is left open in the 2D results in [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Nevertheless, we are able to prove that initial data in little H¨older spaces become smooth for positive times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let X be the solution to the Peskin problem with initial data X0 P h1,γpS2q constructed in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, for any α P p0, 1q, it holds that X P C1pp0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C3,αpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, for any 3 ď n P N and α P p0, 1q, assuming that T P Cn,α, it holds that X P C1pp0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Cn`1,βpS2qq, for any β ă α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We use the solutions constructed in the previous theorem and Duhamel formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) to perform a bootstrapping argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We build on the properties of the semigroup e´tLA (see Section 6 and Appendix B) to first gain regularity in mixed- type spaces Lpp0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Cn,αpS2qq and then transfer this higher regularity in space to show regularity in time as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A key point is that, while the kernels are not of convolution type, we find that it is possible to move derivatives in θ to derivatives in η at the expense of new terms of the same order, but not higher (see (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' As explained above, we must work with the equation localized around a given point and later deal with the corresponding commutators and combine the estimates (see Section 8 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' However, the bootstrapping argument cannot be done on (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) directly, because the right-hand side contains terms of highest regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We combine this process with a regularization argument (see (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14)), where the use of little H¨older spaces becomes crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Related results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The first analytical results for the 2D Peskin problem ap- peared recently in [34,37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In [34], energy arguments are used to prove local well- posedness for H 5 2 initial data and also exponential convergence to steady states for sufficiently close to equilibrium initial data is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The authors in [37] lowered the required initial regularity to barely subcritical spaces, h1,γ, γ P p0, 1q, showed instant smoothing, and provided a blow up criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN After these works, many improvements for the 2D Peskin problem have appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The work [25] deals with the setting in which the enclosed fluid is different to the exterior one, and shows asymptotic stability for small data in Wiener algebra critical spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The result [8] shows the local well-posedness and smoothing for general data in the critical Besov space B 3 2 2,1, including the case of nonlinear elastic law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The sharpest result in terms of regularity appeared in [9], where the semilinear 2D Peskin problem is shown to be well-posed in B1 8,8, and thus with possibly non-Lipschitz curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In relation to the Peskin problem, the article [51] introduces a regularization of the problem inspired by the immersed boundary method and studies its conver- gence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Filaments that resist both bending and stretching are considered in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, we mention two works that introduce simplified models of the 2D Peskin problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The work [23] considers a model for the normal component and shows the existence of global solutions for Lipschitz data near the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Very re- cently, [52] derives a PDE to model the tangential effects of the Peskin problem in the case of an infinitely long and straight string and obtains global solutions with initial data in the energy class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, the author presents many connections of the model with well-known one-dimensional PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' From a mathematical point of view, there are remarkable similarities between the 2D Peskin problem and the so-called Muskat problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In particular, both problems have the same leading linear operator, they can be written in Boundary Integral form [13,30], they have the same scaling and satisfy an energy balance [12, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The Muskat problem, which describes the movement of the interface between incompressible fluids in a porous medium, has been intensively studied in the last two decades [1,2,4,7,12,13,18,36,39], and some of the techniques developed there have been successfully extended in the last years to lower the required regularity for the well-posedness of the 2D Peskin problem [8,9,23,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' However, while there are also results for the 3D Muskat problem [3,5,10,14,24], in all these results the interface is a surface given by graph, hence the geometry does not play a major role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Even in 2D, in the recent non-graph setting [22] that considers a bubble of fluid surrounded by another in a porous medium, a change of parametrization becomes crucial, which is not allowed in the Peskin problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We finally mention some results with more complex elastic interactions [6,11,16, 32,38,42,53,54], mostly dedicated to more qualitative results and weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Part of the interest generated by the Peskin problem is due to its relative simplic- ity, which makes it possible to initiate the analytical study of the rich variety of behaviors in FSI problems, including longtime dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The rest of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In Section 2, we ob- tain the expression for the elastic law and show the Boundary Integral formulation for the 3D Peskin problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 contains the notation used along the paper as well as some definitions and standard results concerning the stereographic pro- jection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, in Section 4, we introduce the operators that will be used later in the paper, we decompose the equation and compute the multiplier of the leading term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Section 5 is dedicated to study the operators previously defined, to show the needed commutators estimates, and to prove Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' These lemmas will be repeatedly used in the proof of the main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In Section 6, we show that the frozen-coefficient operator generates an analytic semigroup (for which we need the WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 9 multiplier results contained in Appendix A, with further properties studied in Ap- pendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, Sections 7 and Section 8 contain the proofs of the main results: Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Formulation and Boundary Integral Reduction The formulation of the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) is closed once that the expression for F el is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' To specify the elastic force F el in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6), we consider the elastic energy of the interface Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We consider an elastic energy EpXq of the form: EpXq “ ż S2 E ˆBX Bθ , θ ˙ dµS2, where µS2 is the standard measure on the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' From this, we may com- pute the elastic force by taking the variational derivative as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let X “ pX1, X2, X3qT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Define the following metric tensors pg and g on S2 and Γ respec- tively, whose i, j components are given by: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) pgij “ Bx X Bθi ¨ Bx X Bθj , gij “ BX Bθi ¨ BX Bθj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We write the energy density as follows: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) E “ AEpsij, θq, sij “ BXi Bθj , i “ 1, ¨ ¨ ¨ 3, j “ 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let Y “ pY1, Y2, Y3qT be a perturbation of the configuration that is compactly supported on the open set U on which the coordinate system θ is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We have: d dτ EpX ` τY q ˇˇˇˇ τ“0 “ ż U BAE Bsij BYi Bθj a detpgdθ1dθ2 “ ´ ż U B Bθj ˆBAE Bsij a detpg ˙ Yidθ1dθ2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) where the summation convention is in effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We set: Fel,i “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='detg B Bθj ˆBAE Bsij a detpg ˙ , where Fel,i are the components of the elastic force F el of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' With this prescription of the elastic force, the solutions satisfy the following energy relation: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) dE dt “ ´ ˜ż Ω 2µ |∇Su|2 dx ` ż R3zΩ 2 |∇Su|2 ¸ dx, ∇Su “ 1 2 ` ∇u ` p∇uqT˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will now impose symmetry conditions to determine the explicit form of AE and hence E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let θ be a (local) orthogonal coordinate system on S2 so that the two coordinate tangent vectors are orthogonal: Bx X Bθ1 ¨ Bx X Bθ2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We thus have an orthonormal frame on (a neighborhood of) S2 given by the two vectors: pei “ Bx X Bθi ����� Bx X Bθi ����� ´1 , i “ 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 10 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN The deformation map X maps the above unit orthogonal vectors to the following two vectors: ei “ BX Bθi ����� Bx X Bθi ����� ´1 , i “ 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Consider the matrix 3 ˆ 2 matrix B “ pe1, e2q whose column vectors are given by ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We may say that the energy density E is a function of B and θ: E “ AEpB, θq, where we have continued to use the notation AE as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By homogeneity of the unit sphere, we impose that AE does not have an explicit dependence on θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Furthermore, the value of AE should not depend on the choice of orthonormal frame pei or the coordinate system in which X resides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This implies the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) AEpBq “ AEpR3BR2q for all R3 P SOp3q and R2 P SOp2q where SOp2q and SOp3q are the group on rotation matrices in 2 and 3 dimensions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let: H “ ˆ e1 ¨ e1 e1 ¨ e2 e1 ¨ e2 e2 ¨ e2 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The invariance condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) implies that AE can only be a function of the trace and determinants of H, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) E “ AEpλ, γq, λ “ a trpHq, γ “ a detpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In terms of the metric tensors g and pg, we can write λ and γ as: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) λ2 “ trppg´1gq, γ2 “ detppg´1gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The above expressions for λ and γ are valid even when θ is not an orthogonal coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We may substitute (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) to obtain: Fel,k “ Fλ,k ` Fγ,k, Fλ,k “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='detg B Bθi ˆ 1 λ BAE Bλ a detpg pgij BXk Bθj ˙ , Fγ,k “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='detg B Bθi ˆBAE Bγ a detg gij BXk Bθj ˙ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) where we use the standard notation aij to denote the inverse tensor pa´1qij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Note that the expressions Fλ,k and Fγ,k are similar but differ crucially in whether pg or g features inside the force expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This is most clearly seen in the following simple cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If we let AE “ λ2{2, we have: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) Fel,k “ Fλ,k “ γ∆S2Xk, ∆S2Xk “ 1 a detpg B Bθi ˆa detpg pgij BXk Bθj ˙ , where ∆S2 is the Laplace-Beltrami operator on the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If we let AE “ γ, we have: Fel,k “ Fγ,k “ ∆ΓXk “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='detg B Bθi ˆa detg gij BXk Bθj ˙ “ ´2κΓnk, where ∆Γ is the Laplace-Beltrami operator of the closed elastic surface Γ, κΓ is the mean curvature of Γ and nk is the k-th component of the outward normal vector n of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This is just the well-known statement on the variation of surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We see from the above expressions that the Fλ,k expresses an elastic force that depends WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 11 strongly on the stretching of the spherical reference configuration whereas Fγ,k is a surface tension force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The prescription of interfacial elastic energy density as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) has its origins the classical work of [19], and may be called the membrane neo-Hookean model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Specific forms for this energy have been used extensively in the modeling and simulation of fluid-structure interaction problems [20,28,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We now rewrite our evolution equation in a form suitable for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence- forth we focus on the case when AE is only a function of λ, and the viscosity µ of the interior fluid is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let us rewrite the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let G be the Stokeslet tensor in R3: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) Gi,jpxq “ 1 8π ˜ δi,j |x| ` xixj |x|3 ¸ , x “ px1, x2, x3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let pFel,k “ 1 γ Fel,k “ 1 a detpg B Bθi ˆ λ´1T pλq a detpg pgij BXk Bθj ˙ “ ∇S2 ¨ ` λ´1T pλq∇S2Xk ˘ , where T pλq “ BAE{Bλ (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let pF el “ pFel,1, Fel,2, Fel,3qT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Notice that we can write λ in terms of X, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) λppxq2 “ |∇S2Xppxq|2, which can be seen from their definitions λ2 “ trppg´1gq “ pgijgji, |∇S2X|2 “ ∇S2Xk ¨ ∇S2Xk “ BXk Bxi BXk Bxl pgijpglmpgjm “ gilpgil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' When µ “ 1, we may write the evolution of X as BX Bt ppxq “ ż S2 GpXppxq ´ XppyqqpF elppyqdµS2ppyq “ ż S2GpXppxq ´ Xppyqq∇S2 ¨ ´ T p|∇S2Xppyq|q ∇S2Xppyq |∇S2Xppyq| ¯ dµS2ppyq, and integrating by parts, we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) BX Bt ppxq“´p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ż S2∇S2GpXppxq´Xppyqq¨T p|∇S2Xppyq|q ∇S2Xppyq |∇S2Xppyq|dµS2ppyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In the following, we will suppress the principal value notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Introducing a smooth partition of unity tρnu, subordinate to a finite atlas of the sphere tUnu, we may write our problem as follows BX Bt ppxq“´ ÿ n ż S2∇S2GpXppxq´Xppyqq¨ T p|∇S2Xppyq|q |∇S2Xppyq| ∇S2` ρnppyqXppyq ˘ dµS2ppyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This can be rewritten using the local charts: BX Bt ppxq “ ´ ÿ n ż Un pgijpηq B Bηi GpXppxq´Xnpηqq ˆ pgjmpηqT p a pgqrgrqpηqq a pgqrgrqpηq pgpmpηq B Bηp ` ρnpηqXnpηq ˘a detpgpηqdη1dη2, 12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN where Xnpηq is the coordinate map on the n-th coordinate chart and ρnpηq “ ρnpx Xpηqq (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 for details of notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We may take an isothermal coordinate system (the stereographic projection gives such a system, for example) on each chart Un, which yields: BX Bt ppxq “ ´ ÿ n ż Un B Bηi GpXppxq´XnpηqqT pλnpηqq λnpηq B Bηi ` ρnpηqXnpηq ˘ dη1dη2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) where we denote (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) λnpηq “ a trppg´1pηqgpηqq “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2}∇Xnpηq}F }∇x Xnpηq}´1 F , and }A}F :“ a trpAT Aq is the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Preliminaries In this section we introduce the notations that will be used in the rest of the paper and summarize some standard results about stereographic projection charts for the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Einstein notation over repeated indices will be of constant use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given vectors v, w and matrices A, B, C with the same size, we denote |v| :“ ∥v∥ “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='vivi, ∥A∥ :“ sup |v|ą0 |Av| |v| “ sup |v|“1 |Av| , A : B :“tr ` ATB ˘ “ AijBij, |A| :“ ∥A∥F :“ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A : A, v b w :“vwT , pA b Bqijkl :“AijBkl, ppA b Bq Cqij :“AijBklCkl “ pB : Cq Aij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will denote C‚,j to the vector given by the jth column of C, and Cj,‚ to the one given by the jth row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will denote µS2 the standard measure on the unit sphere, and for simplicity we will write dpy instead of dµS2ppyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will write high partial derivatives in Rk by multi-index α, where multi-index α is a sequence of k nonegative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' α “ pα1, α2, ¨ ¨ ¨ , αkq P Nk 0, where N0 “ t0u Ş N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 (Multi-index).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given α, β P Nk 0, we have the following arithmetic about the multi-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (i) |α| “α1 ` α2 ` ¨ ¨ ¨ ` αk α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' “α1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='α2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' αk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' α ` β “ pα1 ` β1, α2 ` β2, ¨ ¨ ¨ , αk ` βkq WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 13 (ii) We set α ď β, which is αi ď βi for all i “ 1, 2, ¨ ¨ ¨ , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we have α ´ β “ pα1 ´ β1, α2 ´ β2, ¨ ¨ ¨ , αk ´ βkq ˆ α β ˙ “ α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pα ´ βq!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' “ α1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pα1 ´ β1q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='β1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ¨ ¨ ¨ αk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pαk ´ βkq!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='βk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' High partial derivatives can be written as Bα x f pxq :“ Bα1 Bxα1 1 Bα2 Bxα2 2 ¨ ¨ ¨ Bαk Bxαk k f pxq (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) where α :“ pα1, α2, ¨ ¨ ¨ , αkq and |α| “ α1 ` ¨ ¨ ¨ ` αk is the total number of deriva- tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will use the following set of non-singular matrices (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) DAσ1,σ2 :“ tA : @ξ ‰ 0, σ2 |ξ| ď |Aξ| ď σ1 |ξ|u where σ1 ě σ2 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Euclidean balls of radius R centered at px P Rn will be denoted by Bpx,R, and for balls centered at the origin we will also denote BpRq :“ B0,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will denote Xppx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' tq : S2 ÞÑ Γptq the deformation map that describes the evolving membrane, and we will omit the dependence on time for simplicity of notation, Xppxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will consider a finite atlas tUn, x Xnu of the sphere with 0 P Un, such that the coordinate functions x Xnpθq : Un Ă R2 Ťt8u ÞÑ S2 satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) Bx Xnpθq Bθ1 ¨ Bx Xnpθq Bθ2 “ 0, ����� Bx Xnpθq Bθ1 ����� “ ����� Bx Xnpθq Bθ2 ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In particular, we will choose the standard stereographic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We set tρnu to be a smooth partition of unity subordinate to the coordinate patches tUnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For convenience in the definition of H¨older continuity, we take our system tUn, x Xn, ρnu satisfying the following properties with some R, δ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2 (System tUn, x Xn, ρnu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given R ą 2δ ą 0, we set our isothermal coordinate charts tUnu with the coordinate functions tx Xnpθqu and the partition tρnu to have the following properties: i) Set pxn “ x Xn p0q, then S2 Ă Ť n Bpxn,R, and there exists 0 ă Rn ă 8 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t Bpxn,4R X S2 Ă x Xn pB0,Rnq Ă x XnpUnq ii) @px P S2, Dn s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' x Xn pθq “ px for some θ P B0,Rn, and pBpx,2δ X S2q Ă x Xn pB0,Rnq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' iii) 0 ď ρn ď 1, ρn P C8pS2q, supp pρnq Ă Bpxn,2R X S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' iv) @px P S2, ř n ρn ppxq “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If |x Xn pθq´ x Xn pηq | ě C |θ ´ η| on Un, then Un is totally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given f ppxq : S2 Ñ R, we will denote fn pθq : Un Ă R2 Ñ R with fnpθq :“ fpx Xnpθqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Analogously, we will denote Xnpθq “ Xpx Xnpθqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If x Xn pθq “ x Xm pηq, then Xn pθq “ Xm pηq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 14 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5 (H¨older semi-norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' �f ppxq�Cγ δ pS2q :“ sup 0ă|px´py|ăδ |f ppxq´f ppyq| |px ´ py|γ “ sup n sup 0ă|x Xnpθq´x Xnpηq|ăδ |fn pθq ´ fn pηq| |x Xn pθq´x Xn pηq |γ , �f ppxq�CγpS2q :“ sup 0ă|px´py| |f ppxq ´ f ppyq| |px ´ py|γ “ sup x Xnpθq‰x Xmpηq |fn pθq ´ fm pηq| |x Xn pθq ´ x Xm pηq |γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6 (Arc-chord condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' |f|˚ :“ inf px‰py |f ppxq ´ f ppyq| |px ´ py| “ inf x Xnpθq‰x Xmpηq |fn pθq ´ fm pηq| |x Xn pθq ´ x Xm pηq | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7 (Locally Arc-chord condition in the Charts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given Vn Ă Un, |f|˝,n :“ inf θ‰η,θ,ηPVn |fnpθq ´ fnpηq| |θ ´ η| , |f|˝ :“ inf n |f|˝,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8 (Lp norms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ∥f ppxq∥p LppS2q :“ ÿ n ż Un ρn pθq |fn pθq|p a det pgndθ “ ÿ n ���pρnq 1 p fn ��� p LppUnq ď ÿ n ∥fn pθq∥p LppUnq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Standard Stereographic Projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will see the properties of the standard stereographic projection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' the projection point is p0, 0, 1q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the other projection points, because S2 is centrosymmetric, we just need to rotate the coordinates of S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, most properties among the projection charts are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9 (Standard Stereographic Projection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We set x X : R2 Ťt8u Ñ S2 with x X pθq “ ˜ 2θ1 1 ` |θ|2 , 2θ2 1 ` |θ|2 , ´1 ` |θ|2 1 ` |θ|2 ¸ , x X p8q :“ lim |θ|Ñ8 x X pθq “ p0, 0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, θ ppxq “ ˆ px1 1 ´ px3 , px2 1 ´ px3 ˙ , θ p0, 0, 1q “ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We call the parameterization x X the standard stereographic projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will denote VR the coordinate balls in R2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) VR :“ B ´ R ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 4 ´ R2 ¯ Ă R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The standard stereographic projection has the following prop- erties: WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 15 i) Bx Xpθq Bθ1 ¨ Bx Xpθq Bθ2 “ 0, ����� Bx Xpθq Bθ1 ����� “ ����� Bx Xpθq Bθ2 ����� “ 2 1 ` |θ|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ii) For all R ą 0 and all VR (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4), x XpVRq “ Bp0,0,´1q,R X S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' iii) For θ, η P R2, |x X pθq ´ x X pηq | ď 2 |θ ´ η| , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) and if θ, η P VR with R ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2, |x X pθq ´ x X pηq | ě 2 π |θ ´ η| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For iii), set pξ “ θ´η |θ´η|, then |x X pθq´x X pηq | “ ˇˇˇ ż |θ´η| 0 B Bs x Xpη`spξqds ˇˇˇ ď ż |θ´η| 0 |∇x Xpη`spξq ¨ pξ|ds “ ż |θ´η| 0 2 1`|η ` spξ|2 ds ď 2 |θ ´ η| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Set distppx, py;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' S2q as the length of shortest curve connecting px and py on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' When θ, η P VR with R ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2, the shortest curve ℓ for distpx X pθq , x X pηq ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' S2q is on Bp0,0,´1q,R X S2 and 2 π ď R ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 4 ´ R2 2 cos´1 2´R2 2 ď |x X pθq ´ x X pηq | distpx X pθq , x X pηq ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' S2q ď 1, where the above function of R is a decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, since x X is isothermal and x X ´1 pℓq is in VR, distpx X pθq , x X pηq ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' S2q “ ż ℓ dl ppxq “ ż x X ´1pℓq 2 1 ` |θ|2 dl pθq ě4 ´ R2 2 ż x X ´1pℓq dl pθq ě 4 ´ R2 2 |θ ´ η| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore |x X pθq ´ x X pηq | ě 2 π dist ´ x X pθq , x X pηq ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' S2¯ ě 2 π |θ ´ η| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The Quantitative Relationships between S2 and R2 on the Standard Stereographic Projection Chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given a function f ppxq on S2, we may define f pθq :“ fpx X pθqq on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' x X pθq is isothermal, but the chart is neither isometric nor area-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, some quantities of f between S2 and R2 are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We have to check their quantitative relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' First, let see the H¨older continuous seminorm Cγ and the arc-chord condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given f on S2, it holds that �f�CγpR2q ď 2γ�f�CγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 16 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' �f�CγpR2q “ sup θ‰η ´|x X pθq ´ x X pηq | |θ ´ η| ¯γ |fpx X pθqq ´ fpx X pηqq| |x X pθq ´ x X pηq |γ ď 2γ�f�CγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Notice that |f|˝ for R2 is zero, and the other sided inequality between Cγ pUq and Cγ pUq cannot hold with some constant C ą 0, thus we only can find local inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Set BR “ Bp0,0,´1q,R X S2, VR as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4), and ρ smooth on S2 and supported in BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given R ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2, it holds that �ρf�CγpS2q ď `π 2 ˘γ�ρf�CγpR2q, |f|˚ ďπ 2 |f|˝ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For �ρf�CγpS2q, when px “ x X pθq , py “ x X pηq P BR, θ, η P VR, so |ρf ppxq ´ ρf ppyq| |px ´ py|γ “ ´ |θ ´ η| |x X pθq ´ x X pηq | ¯γ |ρf pθq ´ ρf pηq| |θ ´ η|γ ď `π 2 ˘γ�ρf�CγpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, when px “ x X pθq , py “ x X pηq P Bc R, |ρfppxq´ρfp pyq| |px´ py|γ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, when px “ x X pθq P BR, py “ x X pηq P Bc R, θ P VR, η P V c R, set pz “ x X pξq P BBR s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' |px ´ pz| “ dist ppx, BBRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since ρ is smooth on S2 and supported in BR, ρ ppzq “ 0 “ ρ ppyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, |θ ´ ξ| ď π 2 |px ´ pz| ď π 2 |px ´ py| , so |ρf ppxq ´ ρf ppyq| |px ´ py|γ “ |ρf ppxq ´ ρf ppzq| |px ´ py|γ “ ´ |θ ´ ξ| |x X pθq ´ x X pηq | ¯γ |ρf pθq ´ ρf pξq| |θ ´ ξ|γ ď `π 2 ˘γ�ρf�CγpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Now, for |f|˝, |f|˝ “ inf θ‰η,θ,ηPV |x X pθq ´ x X pηq | |θ ´ η| |fpx X pθqq ´ fpx X pηqq| |x X pθq ´ x X pηq | ě 2 π |f|˚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Next, let us discuss the relationship between C1,γ ` S2˘ and C1,γ ` R2˘ on the standard stereographic projection chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In the standard stereographic projection chart px “ x X pθq, the surface gradient of f, ∇S2f ppxq, is ∇S2f ppxq “ ÿ i,j pgi,j B Bθi f pθq B Bθj x X pθq “ ´1 ` |θ|2 2 ¯2 ÿ i B Bθi f pθq B Bθi x X pθq , where pgij denotes the inverse tensor of pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, 2 1 ` |θ|2 ���∇S2fpx Xpθqq ��� “ |∇f pθq| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We may use the above expressions to obtain the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 17 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' There exist R0 ą 0 and C ě 21`γ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' for all R ă R0 ∥f∥C1,γpR2q ďC ∥f∥C1,γpS2q , ∥ρf∥C1,γpS2q ď ∥ρf∥C1,γpR2q , |f|˚ ď |f|˝ , where ρ is smooth on S2 and supported in BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Stereographic Projection Charts x Xn Covering S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We consider a finite cover of the sphere consisting of balls of radius R ă R0 and center pxn, tBpxn,RXS2u, and a smooth partition of unity subordinated to it, tρnu, such that ρnppxq “ 0 if |px´ pxn| ě 2R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For convenience, we set px0 “ p0, 0, ´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then we take stereographic projection charts x Xn covering S2 (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Stereographic projection charts, x Xnpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14 (Stereographic Projection Charts covering S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' x Xn are standard stereographic projection charts with x Xn : R2 Ñ S2 x Xnpθq “ Θnx Xpθq + , where Θn is the rotation matrix with pxn “ Θnpx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In each chart x Xn, given R ă R0 4 and R0, C from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13, since ρ are supported in Bn, ∥f∥C1,γpR2q ď C ∥f∥C1,γpS2q , ∥ρnf∥C1,γpS2q ď ∥ρnf∥C1,γpR2q , |f|˚ ď |f|˝,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 18 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN As a consequence, |f|˚ ď |f|˝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Nonlinear decomposition In this section we extract the leading structure of the equation and compute its symbol, introducing the notation for the operators that will appear along the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Nonlinear decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will usually consider separately the two terms involved in the Stokeslet kernel (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) Gk,lpxq “ G1 k,lpxq ` G2 k,lpxq, G1 k,lpxq “ 1 8π δk,l |x| , G2 k,lpxq “ 1 8π xkxl |x|3 , and thus we will write our equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) as follows: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) BX Bt ppxq “ FpXqppxq “ F 1pXqppxq ` F 2pXqppxq, where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) FpXqppxq “ ´ ż S2 ∇S2GpXppxq ´ Xppyqq ¨ T p|∇S2Xppyq|q∇S2Xppyqdpy, F jpXqppxq “ ´ ż S2 ∇S2GjpXppxq ´ Xppyqq ¨ T p|∇S2Xppyq|q∇S2Xppyqdpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' and we introduced the notation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) T p|∇S2X|q “ T p|∇S2X|q |∇S2X| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Above, we use the shorter notation dpy “ dµS2ppyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We define the following associate linear operators, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) pNpXqZqkppxq “ ´ ż S2 ∇S2GklpXppxq ´ Xppyqq ¨ Zl,‚ppyqdpy, pN jpXqZqkppxq “ ´ ż S2 ∇S2Gj klpXppxq ´ Xppyqq ¨ Zl,‚ppyqdpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we compute the kernels: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) B Bxi G1 k,lpxq “ ´1 8π xi |x|3 δk,l, B Bxi G2 k,lpxq “ 1 8π δk,ixl ` xkδi,l |x|3 ´ 3 8π xkxlxi |x|5 , and by the chain rule, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) qj k,lppx, pyq : “ ∇S2Gj k,lpXppxq ´ Xppyqq “ ´ B Bxi Gj k,lpXppxq ´ Xppyqq∇S2Xippyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' so that we write (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) pN jpXqZqkppxq “ ´ ż S2 qj k,lppx, pyq ¨ Zl,‚ppyqdpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 19 The explicit expression for qj k,l is given by q1 k,lppx, pyq “ 1 8π ∆ pyXjppxq∇S2Xjppyq |∆ pyXppxq|3 δk,l |px ´ py|2 , and q2 k,lppx, pyq “ ´ 1 8π ∆ pyXlppxq∇S2Xkppyq ` ∆ pyXkppxq∇S2Xlppyq |∆ pyXppxq|3 1 |px ´ py|2 ` 3 8π ∆ pyXkppxq∆ pyXlppxq∆ pyXjppxq∇S2Xjppyq |∆ pyXppxq|5 1 |px ´ py|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Using the standard stereographic projection (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) and the notation Xpθq “ Xpx Xpθqq, the equation for each component of Xpθq becomes BXk Bt pθq “ pFpXqqkpθq “ ´ ż R2 B Bηi Gk,lpXpθq´XpηqqT pλpηqqBXl Bηi pηqdη1dη2, where λpηq is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) and we denote accordingly F jpXqpθq, N jpXqZpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If we use the stereographic projection centered at pxn, then we denote N jpXqZnpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we take the derivative in Gk,l (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) to obtain B Bηi Gk,lpXpθq´Xpηqq “ qi,k,lpθ, ηq “ q1 i,k,lpθ, ηq ` q2 i,k,lpθ, ηq, where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) q1 i,k,lpθ, ηq “ B Bηi G1 k,lpXpθq´Xpηqq “ 1 8πδk,l δηXpθq ¨ BX Bηi pηq |δηXpθq|3 , q2 i,k,lpθ, ηq “ B Bηi G2 k,lpXpθq´Xpηqq “ ´ 1 8π BXk Bηi pηqδηXlpθq ` δηXkpθq BXl Bηi |δηXpθq|3 ` 3 8π δηXkpθqδηXlpθq |δηXpθq|5 δηXpθq ¨ BXpηq Bηi , so that we can write (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) pN jpXqZqkpθq “ ´ ż R2 qj m,k,lpθ, ηq B p Xi Bηm pηqZl,ipηqdη1dη2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We notice that the kernels in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) are given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) qj i,k,lpθ, ηq “ ´ B Bxm Gj k,lpXpθq ´ XpηqqBXm Bηi pηq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We introduce the following notation for finite differences, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) δηgpθq “ gpθq ´ gpηq, ∆ηgpθq “ δηgpθq |θ ´ η|, and we extract the expected leading terms by replacing δηXpθq « ∇Xpηqpθ ´ ηq, p∇Xqp,qpηq “ BXn,p Bηq pηq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 20 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Hence, we define the associate kernels (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) mi,k,lpθ, ηq “ ´ B Bxj Gk,l p∇Xpηqpθ ´ ηqq BXj Bηi pηq “ m1 i,k,lpθ, ηq ` m2 i,k,lpθ, ηq, and define the linear operators Mp∇Xqz as follows: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) pMp∇XqZqkpθq “ ´ ż R2 mm,k,lpθ, ηq B p Xi Bηm pηqZl,ipηqdη1dη2 “ pM1p∇XqZqkpθq ` pM2p∇XqZqkpθq, with pMjp∇XqZqkpθq “ ´ ż R2 mj m,k,lpθ, ηq B p Xi Bηm pηqZl,ipηqdη1dη2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We compute the explicit expression of these kernels mj i,k,l (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13), m1 i,k,lpθ, ηq “ 1 8π BXpηq Bηi ¨ p∇Xpηqpθ´ηqq |∇Xpηqpθ´ηq|3 , m2 i,k,lpθ, ηq “ ´ 1 8π BXkpηq Bηi p∇Xpηqpθ´ηqql ` BXlpηq Bηi p∇Xpηqpθ´ηqqk |∇Xpηqpθ´ηq|3 ` 3 8π p∇Xpηqpθ´ηqqkp∇Xpηqpθ´ηqql |∇Xpηqpθ´ηq|5 p∇Xpηqpθ´ηqq ¨ BXpηq Bηi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will use the notation ∇Xpηqpθ ´ ηq “ pθ ´ ηq ¨ ∇Xpηq, pz “ z |z|, and we define (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15) EηXpθq :“ p{ θ ´ ηq ¨ ∇Xpηq ´ ∆ηXpθq, for which we have that, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16) |EηpXpθqq| |θ ´ η|γ ď �∇X�CγpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, we can write (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='17) NpXqZpθq “ Mp∇XqZpθq ` RpXqZpθq, where the remainder term RpXqZpθq “ 2ÿ j“1 RjpXqZpθq is given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18) pRjpXqZqqkpθq “ pN jpXqZqkpθq ´ pMjp∇XqpZqqkpθq “ ż R2 Kj m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lpθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq B p Xi Bηm pηqZl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ipηqdη1dη2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 21 with kernels K1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lpθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq “ ´q1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='l pθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq ` m1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='l pθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq “ 1 8π δk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='l |θ ´ η|2 BXpηq Bηi ¨ ˆ EηXpθq |∆ηXpθq|3 ´ pp{ θ ´ ηq ¨ ∇Xpηqq ´ 1 |∆ηXpθq|3 ´ 1 |p{ θ ´ ηq ¨ ∇Xpηq|3 ¯˙ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' and K2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lpθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq “ ´q2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='l pθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq ` m2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='l pθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq “ K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lpθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq ` K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lpθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lpθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq “ 1 8π 1 |θ ´ η|2 ˆ ´ BXkpηq Bηi |∆ηXpθq|3 EηXlpθq ` BXkpηq Bηi pp{ θ ´ ηq ¨ ∇Xpηqql ´ 1 |∆ηXpθq|3 ´ 1 |p{ θ ´ ηq ¨ ∇Xpηq|3 ¯ ´ BXlpηq Bηi |∆ηXpθq|3 EηXkpθq ` BXlpηq Bηi pp{ θ ´ ηq ¨ ∇Xpηqqk ´ 1 |∆ηXpθq|3 ´ 1 |p{ θ ´ ηq ¨ ∇Xpηq|3 ¯˙ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lpθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq “ 1 8π 3 |θ ´ η|2 ˆ∆ηXkpθq∆ηXlpθq |∆ηXpθq|5 BXpηq Bηi ¨ EηXpθq ` BXpηq Bηi ¨ pp{ θ ´ ηq ¨ ∇Xpηqq |∆ηXpθq|5 ∆ηXkpθqEηXlpθq ` BXpηq Bηi ¨ pp{ θ ´ ηq ¨ ∇Xpηqq |∆ηXpθq|5 pp{ θ ´ ηq ¨ ∇XpηqqlEηXkpθq ´ BXpηq Bηi ¨ pp{ θ ´ ηq ¨ ∇Xpηqq |∆ηXpθq|5 pp{ θ ´ ηq ¨ ∇Xpηqqlpp{ θ ´ ηq ¨ ∇Xpηqqk ˆ ´ 1 |∆ηXpθq|5 ´ 1 |p{ θ ´ ηq ¨ ∇Xpηq|5 ¯˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Note that for all positive odd integers k, it holds that 1 |u|k ´ 1 |v|k “ pv ´ uq ¨ pu ` vq |u|k ` |v|k řk i“1 |u|2pi´1q|v|2pk´iq |u|k|v|k (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='19) and ���� 1 |u|k ´ 1 |v|k ���� “ ||v| ´ |u|| řk i“1 |u|i´1 |v|k´i |u|k |v|k ď |v ´ u| řk i“1 |u|i´1 |v|k´i |u|k |v|k (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20) In particular, formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='19) with u “ ∆ηXpθq and v “ p{ θ ´ ηq ¨ ∇Xpηq, together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16), makes it clear that there is an extra cancellation in the kernels of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18), 22 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN In summary, our equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) is given by BX Bt pθq “ FpXqpθq “ NpXqpT p|∇S2X|q∇S2Xqpθq “ Mp∇XqpT p|∇S2X|q∇S2Xqpθq ` RpXqpT p|∇S2X|q∇S2Xqpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Symbol of the leading term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' As a preliminary step towards studying the leading term M (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14), let us consider its frozen-coefficient counterpart, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', re- placing ∇Xpηq by a constant matrix A and letting pg “ I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We start with the case T “ Id, that is, T ” 1: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21) pLL AY qkpθq “ p ˜ MpAq∇Y qkpθq, where we define ˜ MpAq “ ˜ M1pAq ` ˜ M2pAq and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22) p ˜ MjpAqZqkpθq “ ´ ż R2 B Bηi pGj k,lpA pθ ´ ηqqqZl,ipηqdη1dη2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let Fθ be the 2D Fourier transform in θ and ξ “ pξ1, ξ2qT: Fθwpξq “ ż R2 wpθq expp´iθ ¨ ξqdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We now compute the Fourier transform of the function GA: GApθq “ GpAθq “ 1 8π ˜ I |Aθ| ` Aθ b Aθ |Aθ|3 ¸ , where I3 is the 3 ˆ 3 identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given that θ P R2 and A is a 3 ˆ 2 matrix, it is convenient to rewrite GA as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' First, note that: |Aθ|2 “ Aθ ¨ Aθ “ θ ¨ ` ATAθ ˘ “ |Bθ|2 , B “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ATA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Notice that B is a 2 ˆ 2 symmetric positive definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Using this B, we have: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='23) GApθq “ 1 8π ˜ I |Bθ| ` Q ˜ Bθ b Bθ |Bθ|3 ¸ QT ¸ , Q “ AB´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We note that Q is an isometry in the sense that QTQ “ I2 where I2 is the 2 ˆ 2 identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We are now ready to compute the Fourier transform of GA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' First, note that: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24) Fθ ˆ 1 |θ| ˙ “ 2π |ξ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, a simple change of variable yields: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='25) Fθ ˆ 1 |Bθ| ˙ “ 2π detpBq |B´1ξ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, note that: Fθ ˜ θiθj |θ|3 ¸ “ Fθ ˆ θiθj∆θ ˆ 1 |θ| ˙˙ “ B2 BξiBξj ˆ |ξ|2 Fθ ˆ 1 |θ| ˙˙ “ 2π B2 BξiBξj |ξ| “ 2π ˜ δij |ξ| ´ ξiξj |ξ|3 ¸ , WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 23 where ∆θ is the Laplacian in R2, δij is the Kronecker delta and we used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24) in the third equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In matrix notation, the above can be written as: Fθ ˜ θ b θ |θ|3 ¸ “ 2π ˜ I2 |ξ| ´ ξ b ξ |ξ|3 ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Again, by changing variables, we see that: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='26) Fθ ˜ Bθ b Bθ |Bθ|3 ¸ “ 2π detpBq ˜ I2 |B´1ξ| ´ B´1ξ b B´1ξ |B´1ξ|3 ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='26), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='23), we obtain: FθGA “ 1 4detpBq ˜ I ` QQT |B´1ξ| ´ QB´1ξ b QB´1ξ |B´1ξ|3 ¸ “ 1 4 a detpATAq ˜ I ` ApATAq´1AT pξ ¨ pATAq´1ξq1{2 ´ ApATAq´1ξ b ApATAq´1ξ pξ ¨ pATAq´1ξq3{2 ¸ This implies that the Fourier symbol of LL A is given by: LL AY “ ´F´1 ξ LL ApξqFθY , LL Apξq “ |ξ|2 pFθGAq pξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' To better understand the properties of Fourier multiplier LL Apξq, we first note that: QQT ´ QB´1ξ b QB´1ξ |B´1ξ|2 “ Q ˜ I2 ´ B´1ξ b B´1ξ |B´1ξ|2 ¸ QT “ vpξq b vpξq, vpξq “ QRπ{2 B´1ξ |B´1ξ|, Rπ{2 “ ˆ 0 ´1 1 0 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='27) Note that vpξq P R3 is a unit vector, and hence, the above matrix 3 ˆ 3 matrix is an orthogonal projection on to the subspace spanned by vpξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We see that: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='28) LL Apξq “ |ξ|2 4detpBq |B´1ξ| pI ` vpξq b vpξqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It is now immediate that LL Apξq is a symmetric positive definite matrix for each ξ ‰ 0 with eigenvalues: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='29) λ “ µ 4 |ξ|2 and µ 2 |ξ|2 , µ “ 1 detpBq |B´1ξ|, where the eigenspace for µ{2 is spanned by vpξq and the two-dimensional eigenspace of µ{4 is spanned by the orthogonal complement of vpξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We also have: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='30) µ 4 |ξ|2 |w|2 ď w ¨ Lpξqw ď µ 2 |ξ|2 |w|2 for any w P R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In the case of general T , the frozen coefficient linear operator can be obtained by a further linearization of the force function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Consider the expression: T pλτq λτ BpXl ` τYlq Bθi , λτ “ λpX ` τY q 24 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN where λ is here viewed as a function of X through its dependence on g (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Now, d dτ ˆT pλτq λτ BpXl ` τYlq Bθi ˙ˇˇˇˇ τ“0 “ ˆ 1 λ dT dλ ´ T λ2 ˙ dλτ dτ ˇˇˇˇ τ“0 BXl Bθi ` T λ BYl Bθi “ ˆ 1 λ dT dλ ´ T λ2 ˙ 1 λppg´1qm,n BXq Bθm BYq Bθn BXl Bθi ` T λ BYl Bθi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Now, the frozen coefficient approximation amounts to taking pg “ I2, BXl{Bθi “ Al,i and λ “ ∥A∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, d dτ ˆT pλτq λτ BpXl ` τYlq Bθi ˙ˇˇˇˇ τ“0 « pTF pAqqi,l,m,q BYq Bθm , with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='31) TFpAq “ T p∥A∥F q ∥A∥F I2 b I2 ´ ˆT p∥A∥F q ∥A∥F ´ dT dλ p∥A∥F q ˙ A b A ∥A∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, the frozen-coefficient linear operator in the general force case is given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='32) pLAY qkpθq “ ´ ż R2 B Bηi pGk,lpA pθ ´ ηqqqpTF pAq∇Y ql,ipηqdη1dη2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let us now take the Fourier transform of the divergence of the above: Fp∇ ¨ pTF pAq∇Y qqpξq “ ´MApξqFY pξq, where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='33) MApξq “ T p∥A∥F q ∥A∥F ˜ |ξ|2 I ´ Aξ b Aξ ∥A∥2 F ¸ ` dT dλ p∥A∥F qAξ b Aξ ∥A∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Note that, if we set T “ Id, then TF “ Id and the above reduces to Mpξq “ |ξ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, in the general case, the multiplier in LApξq of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='32) becomes: LApξq “ pFθGAq pξqMApξq “ I ` vpξq b vpξq 4detpBq |B´1ξ| ˆT p∥A∥F ∥A∥F ˆ |ξ|2 I ´ Aξ b Aξ ∥A∥F ˙ ` dT dλ p∥A∥F qAξ b Aξ ∥A∥F ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='34) It is not difficult to see that, if T ą 0 and dT {dλ ě 0, then the above is coercive in |ξ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Calculus estimates In this section we include some estimates of the operators that will be frequently used in later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 25 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let X P C1pS2q, such that |X|˚ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, the kernels Gj k,lpxq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) and qj k,lppx, pyq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) satisfy the following bounds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) ���Bα x Gj k,lpxq ��� ď C |x|1`|α| , ���∆y ´ Bα x Gj k,lpxq ¯��� ď C M 1`|α| m3`2|α| , |qj k,lppx, pyq| ď C |∇S2Xppyq| |∆ pyXppxq|2 1 |px ´ py|2 ď C }∇S2X}C0pS2q |X|2˚ 1 |px ´ py|2 , where |α| defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1), M “ max p|x| , |y|q, and m “ min p|x| , |y|q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the sake of completeness, we include a version of the divergence theorem that will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Notice that, following standard convention, we will not explicitly write the principal values elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given a matrix A and a compact set D Ă R2 containing 0, then p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ż Dc ∇GpAηqdη : “ lim LÑ8 ż DcXBpLq ∇GpAηqdη “ ´ ż BD GpAηqn pηq dl pηq , where B pLq Ă R2 is the ball centered at 0 of radius L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In particular, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ż R2 ∇GpAηqdη :“ lim LÑ8,εÑ0 ż BpLqzBpεq ∇GpAηqdη “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since D is compact and contains 0, D Ă B pLq when L is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, by integration by parts ż DcXBpLq ∇GpAηqdη “ ´ ż BD GpAηqn pηq dl pηq ` ż BBpLq GpAηqn pηq dl pηq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since GpAηq is even, the boundary term vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, ż Dc ∇GpAηqdη “ lim LÑ8 ż DcXBpLq ∇GpAηqdη “ ´ ż BD GpAηqn pηq dl pηq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, set D “ B pεq, ż R2 ∇GpAηqdη “ lim LÑ8,εÑ0 ż BpLqXBpεqc ∇GpAηqdη “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let A be a matrix in the set DAσ1,σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, the linear operator ˜ MpAq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22) maps CγpR2q X L2pR2q to CγpR2q X L2pR2q for any any γ P p0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, } ˜ MjpAqZ}CγpR2q ď C σ2 ´ 1 ` ´σ1 σ2 ¯2¯ }Z}CγpR2qXL2pR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' And given A1, A2 P DAσ1,σ2 } ˜ MjpA1qZ ´ ˜ MjpA2qZ}CγpR2q ď C σ2 2 ´ 1 ` ´σ1 σ2 ¯5¯ }Z}CγpR2qXL2pR2q ∥A1 ´ A2∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 26 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Taking into account Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2, we have p ˜ MjpAqZqkpθq “ ´ ż R2 B Bηm Gj k,lpApθ ´ ηqq ` Zl,mpηq ´ Cl,m ˘ dη1dη2, where Cl,m is an arbitrary constant, that we will take to be zero or Zl,mpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, the estimate for |p ˜ MjpAqZqkpθq| follows by splitting the integral in two terms, p ˜ MjpAqZqkpθq “ I1pθq ` I2pθq, with I1pθq “ ´ ż |θ´η|ď1 B Bηm Gj k,lpApθ ´ ηqq ` Zl,mpηq ´ Zl,mpθq ˘ dη1dη2, I2pθq “ ´ ż |θ´η|ě1 B Bηm Gj k,lpApθ ´ ηqqZl,mpηqdη1dη2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since B Bηm Gj k,lpApθ ´ ηqq “ ´ BGj k,l Bxi pApθ ´ ηqqAi,m, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) the kernel bounds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) and the fact that A P DAσ1,σ2 provide that |I1pθq| ď C σ1 σ2 2 �Z�CγpR2q, |I2pθq| ď C σ1 σ2 2 }Z}L2pR2q, hence |p ˜ MjpAqZqkpθq| ď C σ1 σ2 2 }Z}CγpR2qXL2pR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We proceed with the seminorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let h P R2, |h| ď 1, and perform the following splitting p ˜ MjpAqZqkpθq ´ p ˜ MjpAqZqkpθ ` hq “ J1 ` J2 ` J3 ` J4, where J1 “ ´ ż |θ´η|ď2|h| B Bηm Gj k,lpApθ ´ ηqqδηZl,mpθqdη, J2 “ ż |θ´η|ď2|h| B Bηm Gj k,lpApθ ` h ´ ηqqδηZl,mpθ ` hqdη, J3 “ δθZl,mpθ ` hq ż |θ´η|ą2|h| B Bηm Gj k,lpθ ´ ηqdη, J4 “ ż |θ´η|ą2|h| ´ B Bηm Gj k,l pApθ`h´ηqq´ B Bηm Gj k,lpApθ´ηqq ¯ δηZl,mpθ`hqdη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Absolutely, |J1| ` |J2| ď C σ1 σ2 2 �Z�CγpR2q |h|γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2, J3 “ pZl,mpθ ` hq ´ Zl,mpθqq ż |θ´η|“2|h| Gj k,l pApθ ´ ηqq nmpηqdlpηq, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 27 and so |J3| ď C�Z�CγpR2q |h|γ 1 σ2 ż |θ´η|“2|h| 1 |θ ´ η|dl pηq ď C 1 σ2 �Z�CγpR2q |h|γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, since B Bθp B Bηm Gj k,lpApθ ´ ηqq “ ´ B Bxq B Bxi Gj k,lpApθ ´ ηqqAi,mAq,p, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) it follows that |J4| “ ˇˇˇ ż |θ´η|ą2|h| ż 1 0 hp B Bθp B Bηm Gj k,l pApθ ` sh ´ ηqq δηpZl,mpθ ` hqdsdη ˇˇˇ ď C σ2 1 σ3 2 �Z�CγpR2q |h| ż |θ´η|ą2|h| ż 1 0 |θ ` h ´ η|γ |θ ` sh ´ η|3 dsdη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In the domain where |θ ´ η| ą 2 |h|, it holds that for s P r0, 1s, 1 2 |θ ` h ´ η| ď |θ ` sh ´ η| ď 3 2 |θ ` h ´ η| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, |J4| ď C σ2 1 σ3 2 �Z�CγpR2q |h|γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, we obtain ��� ˜ MjpAqZ ��� CγpRq ď C σ2 ´ 1 ` σ2 1 σ2 2 ¯ }Z}CγpR2qXL2pR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the L2 norm, since ξmFθ ” Gj k,l pAθq ı pξq is bounded by σ1 and σ2, we have ���p ˜ MjpAqZqk ��� L2pRq “ ���ξmFθ ” Gj k,l pAθq ı pξq FrZl,ms pξq ��� L2pRq ď C pσ1, σ2q ∥FrZs∥L2pRq “ C pσ1, σ2q ∥Z∥L2pRq (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) Next, through (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3), we have ���Gj k,lpA1pθ ´ ηqq´Gj k,lpA2pθ ´ ηqq ��� ď C σ1 |pA1´A2q pθ ´ ηq| σ3 2 |θ ´ η|3 ď C σ1 σ3 2 ∥pA1´A2q∥ |θ ´ η| , ˇˇˇ B Bηm Gj k,lpA1pθ´ηqq´ B Bηm Gj k,lpA2pθ´ηqq ˇˇˇ ď ˇˇˇ BGj k,l Bxi pA1pθ´ηqq´ BGj k,l Bxi pA2pθ´ηqq ˇˇˇ|A1,i,m| ` ˇˇˇ BGj k,l Bxi pA2pθ´ηqq pA1,i,m´A2,i,mq ˇˇˇ ď C ´ 1 σ2 2 ` σ3 1 σ5 2 ¯∥pA1 ´ A2q∥ |θ ´ η|2 , 28 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN and ˇˇˇ B Bθp B Bηm Gj k,lpA1pθ´ηqq´ B Bθp B Bηm Gj k,lpA2pθ´ηqq ˇˇˇ ď ˇˇˇ B Bxq B Bxi Gj k,lpA1pθ´ηqq´ B Bxq B Bxi Gj k,lpA2pθ´ηqq ˇˇˇ|A1,i,mA1,q,p| ` ˇˇˇ B Bxq B Bxi Gj k,lpA2pθ´ηqq pA1,i,mA1,q,p´A2,i,mA2,q,pq ˇˇˇ ď C ´σ1 σ3 2 ` σ5 1 σ7 2 ¯∥pA1´A2q∥ |θ ´ η|3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, we obtain } ˜ MjpA1qZ´ ˜ MjpA2qZ}CγpR2q ď C σ2 2 ´ 1` ´σ1 σ2 ¯5¯ }Z}CγpR2qXL2pR2q ∥A1´A2∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ As an immediate consequence of the previous lemma with ˜Zl,m “ Bx Xi Bηm pηqZl,ipηq, we obtain the following lemma for MpAq: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let A be a matrix in the set DAσ1,σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, the linear operators MjpAq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) map CγpS2q to CγpR2q for any any γ P p0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, }MjpAqZ}CγpR2q ď C σ2 ´ 1 ` ´σ1 σ2 ¯2¯ sup l,m } Bx X Bηm pηq ¨ Zl,‚pηq}CγpR2qXL1pR2q ď Cpσ1, σ2q}Z}CγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In most cases, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4 will be used with Z compactly sup- ported and given by a multiple of a gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Notice that in that case, for Z “ λ∇S2X, λ : R2 ÞÑ R, Bx X Bηm pηq ¨ Zl,‚pηq “ λpηq BXl Bηm pηq, and therefore }MjpAqpλ∇S2Xq}CγpR2q ď Cpσ1, σ2q}λ∇X}CγpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let X P C1pS2q such that |X|˚ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, the linear operators N jpXq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) map CγpS2q to CγpS2q for any γ P p0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) }N jpXqZ}CγpS2q ď C |X|˚ ´ 1 ` ´}∇S2X}C0pS2q |X˚| ¯2¯ }Z}CγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We first notice that we can introduce an arbitrary constant (matrix), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) NpXqZppxq “ ´ ż S2 ∇S2GpXppxq ´ Xppyqq ¨ pZppyq ´ Cqdpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will usually take C “ 0 or C “ Zppxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Recalling the kernels (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8), we write the equation for each component pN jpXqZqkppxq “ ´ ż S2 qj k,lppx, pyq ¨ pZl,‚ppyq ´ Cl,‚qdpy, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 29 where Cl,‚ “ 0 or Cl,‚ “ Zl,‚ppxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We first perform the estimate for |NpXqZppxq|, px P S2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) |NpXqZppxq| ď 2ÿ j“1 |N jpXqZppxq|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Using the bound (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) for the kernel, we have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) |NpXqZppxq| ď C }∇S2X}C0pS2q |X|2˚ ż S2 |Zl,‚ppxq ´ Zl,‚ppyq| |px ´ py|2 dpy ď C }∇S2X}C0pS2q |X|2˚ }Z}CγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We proceed to estimate the H¨older seminorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let px, pxh P S2, and denote h “ |px ´ pxh|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We write pN jpXqZqkppxq´pN jpXqZqkppxhq “ ż S2qj k,lppx, pyq ¨ pZl,‚ppxq´Zl,‚ppyqqdpy ´ ż S2qj k,lppxh, pyq¨pZl,‚ppxhq´Zl,‚ppyqqdpy, and perform the following splitting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) pN jpXqZqkppxq ´ pN jpXqZqkppxhq “ I1 ` I2 ` I3 ` I4, where I1 “ ż t|px´ py|ď2huXS2 qj k,lppx, pyq ¨ pZl,‚ppxq ´ Zl,‚ppyqqdpy, I2 “ ´ ż tpx´ py|ď2huXS2 qj k,lppxh, pyq ¨ pZl,‚ppxhq ´ Zl,‚ppyqqdpy, I3 “ pZl,‚ppxq ´ Zl,‚ppxhqq ¨ ż t|px´ py|ě2huXS2 qj k,lppx, pyqdpy, I4 “ ż t|px´ py|ě2huXS2pqj k,lppx, pyq ´ qj k,lppxh, pyqq ¨ pZl,‚ppxhq ´ Zl,‚ppyqqdpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The first two terms are estimated directly (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) |I1| ` |I2| ď C }∇S2X}C0pS2q |X|2˚ }Z}CγpS2qhγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the third, we use that the kernel is a derivative to integrate by parts and obtain that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) |I3| “ |pZl,‚ppxq ´ Zl,‚ppxhqq ¨ ż t|px´ py|“2huXS2 GjpXppxq ´ XppyqqnppyqdlS2ppyq| ď C }Z}CγpS2q |X|˚ |h|γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, we use the mean-value theorem on the kernel to estimate I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Set ℓpsq the shortest path function from pxh to px respect to arc-length s variable, and L “ 30 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN dist ` pxh, px;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' S2˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we have Gj k,lpXppxq´Xppyqq ´ Gj k,lpXppxhq´Xppyqq “ ż L 0 B BsGj k,lpXpℓ psqq´Xppyqqds “ ż L 0 ∇S2Gj k,lpXpℓ psqq´Xppyqq ¨ B Bsℓ psq ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, for qj k,l (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7), |qj k,lppx, pyq ´ qj k,lppxh, pyq| “ ˇˇˇ∇S2 ż L 0 ∇S2Gk,lpXpℓ psqq´Xppyqq ¨ B Bsℓ psq ds ˇˇˇ “ ˇˇˇ∇S2 ż L 0 B Bxi Gk,lpXpℓ psqq´Xppyqq ˆ ∇S2Xi pℓ psqq ¨ B Bsℓ psq ˙ ds ˇˇˇ “ ˇˇˇ ż L 0 B Bxj B Bxi Gk,lpXpℓ psqq´Xppyqq ˆ ∇S2Xi pℓ psqq¨ B Bsℓ psq ˙ ∇S2Xj ppyq ds ˇˇˇ, and recalling (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) we obtain the bound |qj k,lppx, pyq ´ qj k,lppxh, pyq| ďC ∥∇S2X∥2 C0pS2q |X|3 ˚ ż L 0 1 |ℓ psq ´ py|3 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we have that |I4| ď C }∇S2X}2 C0pS2q}Z}CγpS2q |X|3 ˚ ż t|px´ py|ě2huXS2 |pxh´ py|γ ż L 0 ds |ℓ psq´ py|3 dpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We notice that since ℓpsq is the shortest path function from pxh to px on S2, it holds that |ℓ psq ´ px| ď |px ´ pxh|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, |ℓ psq ´ py| ě |px ´ py| ´ |ℓ psq ´ px| ě |px ´ py| ´ |pxh ´ px| ě 1 2 |px ´ py| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In addition, |pxh ´ py| ď |px ´ py| ` |pxh ´ px| ď 3 2 |px ´ py| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, since L ď Ch, we conclude that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) |I4| ď C }∇S2X}2 C0pS2q}Z}CγpS2q |X|3 ˚ h ż tpx´ py|ě2huXS2 dpy |px ´ py|3´γ ď C }∇S2X}2 C0pS2q}Z}CγpS2q |X|3 ˚ hγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Joining the bounds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) back in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9), we conclude that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) rNpXqZsCγpS2q ď C |X|˚ ´ 1 ` ´}∇S2X}C0pS2q |X˚| ¯2¯ }Z}CγpS2q, and, with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7), the same bound holds for the H¨older norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 31 We will need to localize the operators above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For that purpose, let us define the cutoff function pρn, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) pρnppxq “ # 1 if |px ´ pxn| ď 3R, 0 if |px ´ pxn| ě 4R, and recall the partition of unity tρnu based on the points pxn (see Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let X P C1pS2q such that |X|˚ ą 0, NpXq the linear operator defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5), and pρn the cutoff function (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, for Z P C0pS2q compactly supported on Bpxn,2R X S2, it holds that, }p1 ´ pρnqN jpXqZ}C1pS2q ď CpR, |X|˚, }∇S2X}C0pS2qq}Z}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let Ippxq “ p1 ´ pρnppxqqN jpXqZppxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since 1 ´ pρppxq “ 0 when px P Bpxn,3R, let px P S2zBpxn,3R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, recalling the condition on the support of Z, Ippxq“ppρnppxq´1q ż B pxn,2RXS2 ∇S2GpXppxq´Xppyqq¨Zppyqdpy, and using the bound (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) for the kernel, |Ippxq| ď C }∇S2X}C0pS2q |X|2˚ }Z}C0pS2q ż B pxn,2RXS2 |px´ py|´2dpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since we have that |px ´ py| ě R, we obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15) |Ippxq| ď Cp|X|˚, }∇S2X}C0pS2qq}Z}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' To estimate the H¨older seminorm, consider two points px, pxh P S2, h “ |px ´ pxh|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Due to the cut-off function pρn, the only non-trivial case is px, pxh P S2zBpxn,3R: |Ippxhq ´ Ippxq| “ ppρnppxq ´ pρnppxhqq ż B pxn,2RXS2qk,lppx, pyq ¨ Zl,‚ppyqdpy ` p1´pρnppxhqq ż B pxn,2RXS2 ` qk,lppx, pyq´qk,lppxh, pyq ˘ ¨ Zl,‚ppyqdpy “ J1 ` J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The first term is bounded as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15), |J1| ď C}pρn}C1pS2q }∇S2X}C0pS2q |X|2˚ }Z}C0pS2qh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Recalling the expression of qk,l (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7), we can check that, since |px´py| ě R, |pxh´py| ě R, |qk,lppx, pyq ´ qk,lppxh, pyq| ď C }∇S2X}2 C0pS2q |X|3˚R3 h, hence |J2| ď C R }∇S2X}2 C0pS2q |X|3˚ }Z}C0pS2qh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16) }I}C1pS2q ď CpR, |X|˚, }∇S2X}C0pS2q}C1q}Z}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ 32 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN The previous lemma holds analogously for the operators ˜ MpAq: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let A P DAσ1,σ2, ˜ MjpAq the linear operator defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22), and pρn the cutoff function (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, for Z P C0pR2q compactly supported on V2R, it holds that, }p1 ´ pρnq ˜ MjpAqZ}C1pR2q ď CpR, σ1, σ2q}Z}C0pR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let pxn P S2, X P C1pS2q, |X|˚ ą 0, and ρn P C8pBpxn,2R X S2q, R ă 1{10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, the commutator rρn,NpXqsZppxq “ ´ ż S2∇S2GpXppxq´Xppyqq¨ Zppyqpρnppyq´ρnppxqqdpy, satisfies that, for any γ P p0, 1q, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='17) ∥rρn, NpXqsZ∥CγpS2q ď Cp|X˚, }∇S2X}C0pS2qq}∇S2ρn}C0pS2q}Z}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Recalling the kernel bound (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1), |rρn, NpXqsZppxq| ď C }∇S2ρn}C0pS2q}Z}C0pS2q}∇S2X}C0pS2q |X|2˚ ż S2 1 |px ´ py|dpy ď Cp|X˚, }∇S2X|C0pS2qq}∇S2ρn}C0pS2q}Z}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, we study the H¨older seminorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We take two points px, pxh, denote h “ |px ´ pxh|, and perform a splitting analogous to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Using the kernel notation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' rρn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' NpXqsZppxq´rρn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' NpXqsZppxhq“ ż S2 qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lppx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pyq pρnppyq´ρnppxqq¨Zl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='‚ppyqdpy ´ ż S2qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lppxh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pyq pρnppyq´ρnppxhqq¨Zl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='‚ppyqdpy “ I1 n ` I2 n ` I3 n ` I4 n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' where I1 n “ ż t|px´ py|ď2huXS2 qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lppx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pyq pρnppyq ´ ρnppxqq ¨ Zl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='‚ppyqdpy I2 n “ ´ ż t|px´ py|ď2huXS2 qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lppxh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pyq pρnppyq ´ ρnppxhqq ¨ Zl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='‚ppyqdpy I3 n “ ż t|px´ py|ě2huXS2 qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lppx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pyq pρnppxhq ´ ρnppxqq ¨ Zl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='‚ppyqdpy and I4 n “ ż t|px´ py|ě2huXS2 pqk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lppx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pyq´qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lppxh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pyqq pρnppyq´ρnppxhqq ¨ Zl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='‚ppyqdpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, for I1 n and I2 n, |I1 n| ` |I2 n| ď C ∥∇S2ρn∥C0pS2q ∥∇S2X∥C0pS2q |X|2 ˚ }Z}C0pS2qh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 33 Next, for I3 n, integration by parts gives that ��I3 n �� ď C |ρnppxhq´ρnppxq| ż t|px´ py|ě2huXS2 }Z}C0pS2q ∥∇S2X∥C0pS2q |X|2 ˚ |px ´ py|2 dpy ď Cp|X˚, }∇S2X}C0pS2qq}∇S2ρn}C0pS2q}Z}C0pS2qh log h´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, in I4 n, the use of the mean-value theorem provides that ��I4 n �� ď C ∥∇S2ρn∥C0pS2q }Z}C0pS2q ∥∇S2X∥2 C0pS2q |X|3 ˚ ˆ ż t|px´ py|ď2huXS2 ż L 0 |pxh ´ py| |ℓ psq ´ py|3 dsdpy ď Cp|X˚, ∥∇S2X∥C0pS2qq ∥∇S2ρn∥C0pS2q ∥Z∥C0pS2q h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, ∥rρn, NpXqsZ∥CγpS2q ď Cp|X˚, }∇S2X}C0pS2qq}∇S2ρn}C0pS2q}Z}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let pxn P S2 and x Xn : R2 Y t8u Ñ S2 the stereographic projection centered at pxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let X P C1,γpS2q, |X|˚ ą 0, and pρn given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let the linear operators NpXq and MpAq be defined by in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14), with A “ ∇Xnp0q, where we are using the notation Xnpθq “ Xpx Xnpθqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Denote Inpθq “ pρnpθqrMpAq ´ NpXqsZnpθq, Then, for Z P CγpS2q compactly supported on Bpxn,2R X S2, the following estimates hold: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18) }In}CγpR2q ď Cp|X|˚, }∇S2X}C0pS2qq ` p1 ` }∇S2X}CγpB pxn,5RXS2qq}Z}C0pS2q ` εpRq}Z}CγpS2q ˘ , and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='19) }In}CγpR2q ď Cp|X|˚, }∇S2X}C0pS2qq ` }Z}C0pS2q`}∇S2X}C γ 2 pB pxn,5RXS2q}Z}C γ 2 pS2q ` εpRq}Z}CγpS2q ˘ , with εpRq Ñ 0 as R Ñ 0 given by the modulus of continuity of ∇S2X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' First, we write (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20) In “ I1 n ` I2 n “ 2ÿ j“1 pρnpθqrMjpAq ´ N jpXqsZnpθq “ 2ÿ j“1 pρnpθqrMjpAq ´ Mjp∇XnqsZnpθq ´ RjpXnqZnpθq, and focus on the term I1 n, as the other I2 n will follow similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We then write I1 n “ ´pρnpθq ż R2 P 1 m,k,lpθ, ηq B p Xi Bηm pηqZn,lipηqdη1dη2, 34 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN where we used the notation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21) P 1 m,k,lpθ, ηq “δk,l 8π B Bηm ´ 1 |Apθ ´ ηq| ´ 1 |Xnpθq ´ Xnpηq| ¯ “ ´ 1 8π pBpθ´ηqqm |Apθ´ηq|3 δk,l ` q1 m,k,lpθ, ηq “ ´ 1 8π pBpθ´ηqqm |Apθ´ηq|3 δk,l ` m1 m,k,lpθ, ηq ´ K1 m,k,lpθ, ηq “ ´ 1 8π ˆpBpθ´ηqqm |Apθ´ηq|3 ´ pBnpηqpθ´ηqqm |Anpηqpθ´ηq|3 ˙ δk,l ´ K1 m,k,lpθ, ηq :“P1 m,k,lpθ, ηq ´ K1 m,k,lpθ, ηq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Above, we are denoting B “ AT A, A “ ∇Xnp0q, Anpηq “ ∇Xnpηq, and K1 m,k,l was given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We denote ˜Zl,mpηq “ B p Xi Bηm pηqZn,lipηq, and note that } ˜Z}CγpR2q ď C}Z}CγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We start with a bound for |I1 n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We split it as follows (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22) I1 n “ O1 ` O2, with O1 “ pρnpθq ż V5R P 1 mklpθ, ηq ` ˜Zl,mpηq ´ ˜Zl,mpθq ˘ dη1dη2 O2 “ pρnpθq ˜Zl,mpθq ż V5R P 1 m,k,lpθ, ηqdη1dη2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we split O1 further |O1| ď O1,1 ` O1,2, where O1,1 “ } ˜Z}CγpV5Rq ż V5R |P1 m,k,lpθ, ηq||θ ´ η|γdη1dη2, O1,2 “ pρnpθq ż V5R |K1 m,k,lpθ, ηq||δη ˜Zl,mpθq|dη1dη2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For θ P V4R, η P V5R, we have the following bound for P1 m,k,l (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='23) |P1 m,k,l| ď 1 8π |pBnpηq ´ Bqpθ ´ ηq |Apθ ´ ηq|3 | ` 1 8π |Bnpηqpθ ´ ηq|| 1 |Apθ ´ ηq|3 ´ 1 |Anpηqpθ ´ ηq|3 | ď C }Bnp¨q ´ B}C0pV5Rq |X|3˝,n|θ ´ η|2 ` C }Anp¨q ´ A}C0pV5Rq}∇Xn}7 C0pV5Rq |X|9˝,n|θ ´ η|2 , WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 35 where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='19) has been used for the last term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, the first term O1,1 is high-order but with small coefficients, O1,1 ď C ´}Bnp¨q ´ B}C0pV5Rq |X|3˝,n ` }Anp¨q ´ A}C0pV5Rq}∇Xn}7 C0pV5Rq |X|9˝,n ¯ Rγ ˆ } ˜Z}CγpV5Rq, since we have that }Anp¨q ´ A}C0pV5Rq ď εpRqCp}∇X}C0pV5Rqq, with εpRq Ñ 0 as R Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The kernel bound, with θ P V4R, η P V5R, |K1 m,k,lpθ, ηq| ď C }∇Xn}C0pV5Rq |X|3˝,n r∇XnsCγpV5Rq 1 |θ ´ η|2´γ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24) gives that O1,2 ď C }∇Xn}C0pV5Rq}∇Xn}CγpV5Rq |X|3˝,n Rγ} ˜Z}C0pV5Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' But one also has the bound |K1 m,k,lpθ, ηq| ď C }∇Xn}C0pV5Rq}∇Xn}C γ 2 pV5Rq |X|3˝,n 1 |θ ´ η|2´ γ 2 , which gives that O1,2 ď C }∇Xn}C0pV5Rq}∇Xn}C γ 2 |X|3˝,n Rγ} ˜Z}C γ 2 pV5Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Joining the two bounds, we obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='25) |O1| ď Cp|X|˚, }∇S2X}C0pS2qqRγ` } ˜Z}C0pV5Rq}∇S2X}CγpB pxn,5RXS2q ` εpRq} ˜Z}CγpV5Rq ˘ , and |O1| ď Cp|X|˚, }∇S2X}C0pS2qqRγ` }∇S2X}C γ 2 pB pxn,5RXS2q} ˜Z}C γ 2 pV5Rq ` εpRq} ˜Z}CγpV5Rq ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' To estimate O2, we use (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21) to integrate by parts, |O2| ď C|pρnpθq|| ˜Zl,mpθq| ż BV5R ˇˇˇ 1 |Apθ ´ ηq| ´ 1 |Xnpθq ´ Xnpηq| ˇˇˇdlpηq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, we note that since θ P V4R, we have that for η P BV5R, |θ ´ η| ě R 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We compute the difference (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='26) 1 |Apθ ´ ηq| ´ 1 |Xnpθq ´ Xnpηq| “ 1 |θ ´ η| ´ 1 |Ap{ θ ´ ηq| ´ 1 |∆ηXnpθq| ¯ “ 1 |θ´η| ` ∆ηXnpθq´Ap{ θ ´ ηq ˘ ¨ ` ∆ηXnpθq`Apz θ´ηq ˘ |Apz θ´ηq||∆ηXnpθq| ` |Apz θ´ηq|`|∆ηXnpθq| ˘ , 36 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN and ∆ηXnpθq´Ap{ θ ´ ηq “ ∆ηXnpθq´∇Xnpηqp{ θ ´ ηq`p∇Xnpηq ´ Aqp{ θ ´ ηq “ ´EηXnpθq ` pAnpηq ´ Aqp{ θ ´ ηq, where EηXnpθq is given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, |O2| ď C} ˜Z}C0pV5Rq }∇Xn}C0pV5Rq |X|3˝,n ż BV5R |EηXnpθq|`}Anp¨q´A}C0pV5Rq |θ´η| dlpηq, hence |O2| ď Cp|X|˚, }∇S2X}C0pS2qq} ˜Z}C0pV5RqpR γ 2 }∇S2X}C γ 2 pB pxn,5RXS2q ` εpRqq, and |O2| ď Cp|X|˚, }∇S2X}C0pS2qqRγ}∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='25) back in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22), we conclude that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='27) |I1 n| ď Cp|X|˚, }∇S2X}C0pS2qqRγ` } ˜Z}C0pV5Rq}∇S2X}CγpB pxn,5RXS2q ` εpRq} ˜Z}CγpV5Rq ˘ , and also |I1 n| ď Cp|X|˚, }∇S2X}C0pS2qq ` R γ 2 }∇S2X}C γ 2 pB pxn,5RXS2q} ˜Z}C γ 2 pV5Rq ` εpRq} ˜Z}CγpV5Rq ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We proceed to estimate the H¨older seminorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Take θ, θ ` h P V4R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We use the splitting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22), and start with the estimate for O1: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='28) |O1pθ ` hq ´ O1pθq| “ |Q1 ` Q2 ` Q3 ` Q4 ` Q5|, where, using the notation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12), Q1 “ ´pρnpθ ` hq ż t|θ´η|ď2|h|uXV5R P 1 m,k,lpθ ` h, ηqδη ˜Zl,mpθ ` hqdη1dη2, Q2 “ ´pρnpθ ` hq ż t|θ´η|ď2|h|uXV5R P 1 m,k,lpθ, ηqδη ˜Zl,mpθqdη1dη2, Q3 “ pρnpθ ` hqδθ ˜Zl,mpθ ` hq ż t|θ´η|ě2|h|uXV5R P 1 m,k,lpθ, ηqdη1dη2, Q4 “ pρnpθ ` hq ż t|θ´η|ě2|h|uXV5R pP 1 m,k,lpθ, ηq ´ P 1 m,k,lpθ ` h, ηqqδη ˜Zl,mpθ ` hqdη1dη2, Q5 “ ppρnpθq ´ pρnpθ ` hqq ż V5R P 1 m,k,lpθ, ηqδη ˜Zl,mpθqdη1dη2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Recalling (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21) and the bounds for P1 m,k,l (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='23) and K1 m,k,l (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24), we obtain |Q1| ` |Q2| ď C ´}Bnp¨q ´ B}C0pV5Rq |X|3˝,n ` }Anp¨q ´ A}C0pV5Rq}∇Xn}7 C0pV5Rq |X|9˝,n ¯ ˆ ż t|θ´η|ď2|h|uXV5R r ˜ZsCγpV5Rq |θ ´ η|2´γ dη1dη2 ` C }∇Xn}C0pV5Rq |X|3˝,n ż t|θ´η|ď2|h|uXV5R r∇XnsCγpV5Rq} ˜Z}C0pV5Rq |θ ´ η|2´γ dη1dη2, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 37 thus (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='29) |Q1|`|Q2| ď Cp|X|˚, }∇S2X}C0pS2qq ` }∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq ` εpRq} ˜Z}CγpV5Rq ˘ |h|γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It is clear that we could also obtain the estimate |Q1|`|Q2| ď Cp|X|˚, }∇S2X}C0pS2qq|h|γ` }∇S2X}C γ 2 pB pxn,5RXS2q} ˜Z}C γ 2 pV5Rq ` εpRq} ˜Z}CγpV5Rq ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We see that the difference between those two type of estimate comes only from the kernel K, where we can distribute half a derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The same idea propagates along the lines below, hence we only show the first estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, integration by parts in Q3 gives |Q3| ď C|pρnpθ ` hq||δθ ˜Zpθ ` hq| ˆ ż t|θ´η|“2|h|uYBV5R ˇˇˇ 1 |Apθ ´ ηq| ´ 1 |Xnpθq ´ Xnpηq| ˇˇˇdlpηq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='26), |h| ď 8R and that θ P V4R, we obtain that ż t|θ´η|“2|h|uYBV5R ˇˇˇ 1 |Apθ ´ ηq| ´ 1 |Xnpθq ´ Xnpηq| ˇˇˇdlpηq ď C }∇X}C0pV5Rq |X|3˝,n ż t|θ´η|“2|h|uYBV5R |EηXnpθq| |θ´η| ` }Anp¨q´A}C0pV5Rq |θ´η| dlpηq ď C }∇X}C0pV5Rq |X|3˝,n ´ }∇X}C0pV5Rqpεp|h|q ` εpRqq ` }∇X}C0pV5RqεpRq ¯ ď C }∇X}2 C0pV5Rq |X|3˝,n εpRq, hence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='30) |Q3| ď εpRqCp|X|˚, }∇S2X}C0pS2qq} ˜Z}CγpV5Rq|h|γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The term Q4 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='28) is estimated by applying the mean-value theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' As in the previous terms, we need to consider separately the two kernels in P 1 m,k,l (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We take a derivative on P1 m,k,l, B Bθj P1 m,k,lpθ, ηq “ δk,l 8π ´ ´ Bm,j |Apθ ´ ηq|3 ` pBnpηqqm,j |Anpηqpθ ´ ηq|3 ¯ ` δk,l 8π ´ 3pBpθ ´ ηqqmpBpθ ´ ηqqj |Apθ ´ ηq|5 ´ 3pBnpηqpθ ´ ηqqmpBnpηqpθ ´ ηqqj |Anpηqpθ ´ ηq|5 ¯ , and thus | B Bθj P1 m,k,lpθ, ηq| ď Cp}∇Xn}C0pV4Rq, |X|˝,nq εpRq |θ ´ η|3 , 38 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN while the derivative of K1 m,k,l is computed and bounded below B Bθj K1,1 m,k,lpθ, ηq “ ´δk,l BXnpηq Bηm ¨ ˆ ¨ ˝3 ∆ηXnpθq ¨ BXnpθq Bθj |∆ηXnpθq|5 EηXnpθq |θ ´ η|3 ` δη BXnpθq Bθj |∆ηXnpθq|3|θ ´ η|3 ˛ ‚, | B Bθj K1,1pθ, ηq| ď }∇Xn}C0pV4Rq |X|3˝,n r∇XnsCγpV4Rq |θ ´ η|3´γ ˆ 1`3}∇Xn}C0pV4Rq |X|˝,n ˙ ď Cp|X|˚, }∇S2X}C0pS2qq}∇S2X}CγpB pxn,2RXS2q |θ ´ η|3´γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, we obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='31) |Q4| ď Cp|X|˚, }∇S2X}C0pS2qq ` }∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq ` εpRq} ˜Z}CγpV5Rq ˘ |h|γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, the estimate for Q5 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='28) follows from the bounds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='23) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='32) |Q5| ď Cp}∇Xn}C0pV5Rq, |X|˝,nq}pρn}CγpV5Rq|h|γ ˆ ´ εpRq} ˜Z}CγpV5Rq`}∇Xn}CγpV5Rq} ˜Z}C0pV5Rq ¯ ď Cp|X|˚, }∇S2X}C0pS2qq ` }∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq ` εpRq} ˜Z}CγpV5Rq ˘ |h|γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We combine the bounds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='29), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='30), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='31), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='32) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='28) to conclude that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='33) |O1pθ ` hq ´ O1pθq| ď Cp|X|˚, }∇S2X}C0pS2qq ` }∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq ` εpRq} ˜Z}CγpV5Rq ˘ |h|γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We continue with the H¨older seminorm for O2 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Integrating by parts O2pθ ` hq ´ O2pθq “ δk,l 8π pρnpθ ` hq ˜Zl,mpθ ` hq ˆ ż BV5R ´ 1 |Apθ ` h ´ ηq| ´ 1 |Xpθ ` hq ´ Xnpηq| ¯ nmpηqdlpηq ´ δk,l 8π pρnpθq ˜Zl,mpθq ż BV5R ´ 1 |Apθ´ηq| ´ 1 |Xnpθq´Xnpηq| ¯ nmpηqdlpηq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='34) |O2pθ ` hq ´ O2pθq| “ |Q6 ` Q7|, with Q6 “ δk,l 8π ´ pρnpθ ` hq ˜Zl,mpθ ` hq ´ pρnpθq ˜Zl,mpθq ¯ ˆ ż BV5R | 1 |Apθ ` h ´ ηq| ´ 1 |Xpθ ` hq ´ Xnpηq||nmpηqdlpηq, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 39 Q7 “ δk,l 8π pρnpθq ˜Zl,mpθq ˆ ˆ ż BV5R ´ 1 |Apθ ` h ´ ηq| ´ 1 |Xnpθ ` hq ´ Xnpηq| ¯ nmpηqdlpηq ´ ż BV5R ´ 1 |Apθ ´ ηq| ´ 1 |Xnpθq ´ Xnpηq| ¯ nmpηqdlpηq ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='26) and that θ P V4R, we obtain |Q6| ď C} ˜Z}CγpV5Rq|h|γ }∇Xn}C0pV5Rq |X|3˝,n ˆ ż BV5R |EηXnpθq|`}Anp¨q´A}C0pV5Rq |θ´η| dlpηq ď C} ˜Z}CγpV5Rq|h|γ }∇Xn}C0pV5Rq |X|3˝,n }∇Xn}C0pV5RqεpRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, we estimate Q7, |Q7| ď C} ˜Z}C0pV5Rq ż BV5R | 1 |Apθ ` h ´ ηq| ´ 1 |Apθ ´ ηq||dlpηq ` C} ˜Z}C0pV5Rq ż BV5R | 1 |Xnpθ ` hq ´ Xnpηq| ´ 1 |Xnpθq ´ Xnpηq||dlpηq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Performing the differences we obtain that |Q7| ď Cp|X|˝,n, }∇Xn}C0pV5Rqqp|h| ` |h|1´γRγq R } ˜Z}C0pV5Rq|h|γ ď Cp|X|˝,n, }∇Xn}C0pV5Rqq} ˜Z}C0pV5Rq|h|γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, going back to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='34), we conclude that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='35) |O2pθ`hq´O2pθq|ďCp|X|˚, }∇S2X}C0pS2qq ` } ˜Z}C0pV5Rq ` εpRq} ˜Z}CγpV5Rq ˘ |h|γ, and together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='33) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='27) back into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22) we obtain the H¨older norm estimate for I1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since the kernel in I2 n (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20) satisfy the same estimates, we conclude that }In}CγpR2q ď Cp|X|˚, }∇S2X}C0pS2qq ` p1 ` }∇S2X}CγpB pxn,5RXS2qq} ˜Z}C0pV5Rq ` εpRq} ˜Z}CγpV5Rq ˘ |h|γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Frozen-coefficient Semigroup We will need later in the proof (see Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) to deal with the following kernels, with 0 ď α ď 1: GαpAθq “ G1pAθq ` αG2pAθq, 40 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN where G1, G2 are given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' From Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2, we have that pFθGα,Aq pξq “ ` FθG1pAθq ˘ pξq ` α ` FθG2pAθq ˘ pξq, ` FθG1pAθq ˘ pξq “ 1 4 detpBq |Uξ|, ` FθG2pAθq ˘ pξq “ 1 4 detpBq |Uξ| ˆ P ´ Uξ |Uξ| b Uξ |Uξ| ˙ , where λ “ ∥A∥F , B “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ATA, P “ ApATAq´1AT, U “ ApATAq´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let us consider the operator defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In preparation for this, we consider the operator with ∇X given by a constant matrix and with pg “ I2, as defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='32): Lα,AY :“ ˜ Mα pAq pTFpAq∇Y q “ ´ ˜ M1pAq ` α ˜ M2pAq ¯ pTF pAq∇Y q where TFpAq is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The parameter α is useful in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will prove Lα,A is a sectorial operator first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' That is to say, we have to estimate pz ` Lα,Aq´1 where z is in a set with some ω P R, 0 ă δ ă π 2 in the complex plane: Sω,δ “ tz P C : |argpz ´ ωq| ď π ´ δu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since ˜ MjpAq is a singular integral operator, it is difficult to compute its inverse operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' However, ˜ MjpAq is a convolution with kernel Gj pAθq, so we may use its Fourier multiplier to obtain pz ` Lα,Aq´1 Y “F´1ppz ´ Lα,Apξqq´1 FY pξqqpθq where Lα,A is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='34) (in this section, we will write LA instead of Lα,A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we will use the Fourier multiplier to estimate the original operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In harmonic analysis, the Fourier multiplier theorem in Lp norms is well-known [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will use a Fourier multiplier theorem in semi-norms �¨�CγpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If T is a Fourier multiplier operator with multiplier m pξq P Cs pRnzt0uq X L8 pRnq , s ą n 2 , and ��Bα ξ m pξq �� ď Cα |ξ|´|α| for all |α| ď s, then for all u P Cγ pRnq X L2 pRnq where 0 ă γ ă 1, �T u�CγpRnq ď Cγ,s,nDm�u�CγpRnq, where Dm “ max|α|ďs Cα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 may be split into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The first part is the well-known equivalence between the homogeneous Besov norm ∥¨∥ 9Bγ 8,8pRnq and H¨older seminorm �¨�CγpRnq [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The second part is proving the Fourier mul- tiplier theorem in homogeneous Besov norms ∥¨∥ 9Bγ 8,8pRnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Although these results are classical, we include the proof of the version we need in Appendix A for the covenience of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will compute pz ´ Lα,Apξqq´1 in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, for the norm ∥¨∥C0pR2q, we may expand the result in [47, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1] in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 41 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3 ( [47, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If �u�CγpR2q ă 8 for some γ P p0, 1q and Fu pξq “ 0 in a neighborhood of ξ “ 0, then ∥u∥C0pR2q ď C�u�CγpR2q, where C depends on the neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, we may choose a suitable cutting function ϕ pξq with ϕ pξq “ 1 in a neighborhood of ξ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, the rest of the work for ���pz ` Lα,Aq´1 Y ��� C0pR2q is only ���F´1ppz ´ Lα,Apξqq´1 ϕ pξq FY pξqqpθq ��� C0pR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Fundamental estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We need some elementary estimates on operators LA and LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' To achieve it, we first compute the estimate of TFpAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given matrice A1, A2 in DAσ1,σ2, we have σ2 ď ∥A1∥F , ∥A2∥F ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we have the following estimates for TF pAq: |TF pA1qijkl| ďzp0q M , |TFpA1qijkl ´ TF pA2qijkl| ďC ˆ zp0q M σ1 σ2 2 ` zp1q M ˙ ∥A1 ´ A2∥ , where zp0q M “ max σ2ďλď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2σ1 |f1 pλq| ` |f2 pλq| , zp1q M “ max σ2ďλď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2σ1 ���� df1 dλ ���� ` ���� df2 dλ ���� , f1 pλq “ T λ , f2 pλq “ T λ ´ dT dλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' More specific, given Z P Cγ ` R2˘ Ş L2 ` R2˘ with the size of A1, ∥TFpA1qZ∥CγpR2q Ş L2pR2q ďCzp0q M ∥Z∥CγpR2q Ş L2pR2q (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) ∥pTF pA1q ´ TFpA2qq Z∥CγpR2q Ş L2pR2q ďC ˆ zp0q M σ1 σ2 2 ` zp1q M ˙ ∥A1 ´ A2∥ ∥Z∥CγpR2q Ş L2pR2q (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Set λi “ ∥Ai∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since TF pA1qijkl “ f1 pλ1q δikδjl ´ f2 pλ1q pA1qij pA1qkl λ2 1 , It is obvious to obtain the result of TF pAqijkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, TF pA1qijkl ´ TF pA2qijkl “ pf1 pλ1q ´ f1 pλ2qq δikδjl ´ pf2 pλ1q ´ f2 pλ2qq pA1qij pA1qkl λ2 1 ´ f2 pλ2q ˆpA1qij pA1qkl λ2 1 ´ pA2qij pA2qkl λ2 2 ˙ Since |λ1 ´ λ2| ď ∥A1 ´ A2∥F ď C ∥A1 ´ A2∥ , we obtain |fi pλ1q ´ fi pλ2q| ď zp1q M |λ1 ´ λ2| ď Czp1q M ∥A1 ´ A2∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 42 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN pA1qij pA1qkl λ2 1 ´ pA2qij pA2qkl λ2 2 “ pA1 ´ A2qij pA1qkl ` pA2qij pA1 ´ A2qkl λ2 1 ` pA2qij pA2qkl pλ2 ` λ1q pλ2 ´ λ1q λ2 1λ2 2 , so ���� pA1qij pA1qkl λ2 1 ´ pA2qij pA2qkl λ2 2 ���� ď C σ1 σ2 2 ∥A1 ´ A2∥ Therefore, |TF pA1qijkl ´ TF pA2qijkl| ď C ˆ zp0q M σ1 σ2 2 ` zp1q M ˙ ∥A1 ´ A2∥ Since TF pA1q, TF pA2q are linear operators, we may obtain the estimates in Cγ ` R2˘ X L2 ` R2˘ of TFpA1qZ and pTF pA1q ´ TFpA2qq Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Then, because we have estimated ˜ Mα in Thoerem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3, we can obtain the bounds of Lα,A and its difference Lα,A1 ´ Lα,A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given a matrix A1, A2 P DAσ1,σ2, then for all Y P C1,γ ` R2˘ compactly supported, Lα,AY P Cγ ` R2˘ X L2 ` R2˘ , and ∥Lα,AY ∥CγpRq ď zp0q M σ2 ´ 1 ` ´σ1 σ2 ¯2¯ ∥∇Y ∥CγpR2q Ş L2pR2q , ∥Lα,A1Y ´ Lα,A2Y ∥CγpRq ď C σ2 ´zp0q M σ2 ´ 1 ` σ1 σ2 ¯ ` zp1q M ¯´ 1 ` ´σ1 σ2 ¯5¯ ∥∇Y ∥CγpR2q Ş L2pR2q ∥A1 ´ A2∥ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given Y P C1,γ ` R2˘ compactly supported, ∇Y is also in L2 ` R2˘ , so pTF pAq∇Y q P Cγ ` R2˘ X L2 ` R2˘ by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4, Lα,AY P Cγ ` R2˘ X L2 ` R2˘ and ∥Lα,AY ∥CγpRq ď C σ2 ´ 1 ` ´σ1 σ2 ¯2¯ ∥TF pAq∇Y ∥CγpR2q Ş L2pR2q ďzp0q M σ2 ´ 1 ` ´σ1 σ2 ¯2¯ ∥∇Y ∥CγpR2q Ş L2pR2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, since Lα,A1Y ´ Lα,A2Y “ ´ ˜ Mα pA1q ´ ˜ Mα pA2q ¯ pTF pA1q∇Y q ´ ˜ Mα pA2q ppTF pA1q ´ TF pA2qq ∇Y q , by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4, ∥Lα,A1Y ´ Lα,A2Y ∥CγpRq ď C zp0q M σ2 2 ´ 1 ` ´σ1 σ2 ¯5¯ ∥∇Y ∥CγpR2q Ş L2pR2q ∥A1 ´ A2∥ ` C σ2 ˆ zp0q M σ1 σ2 2 ` zp1q M ˙ ´ 1 ` ´σ1 σ2 ¯2¯ ∥∇Y ∥CγpR2q Ş L2pR2q ∥A1 ´ A2∥ □ WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 43 Next, we compute some elementary estimates on the symbol LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We denote Gα,A pθq :“ G1 pAθq ` αG2 pAθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The pairs of eigenvalues and respective eigenvec- tors of pFθGα,Aq pξq are ´ p1 ` αq µ 4 , v pξq ¯ , ´µ 4 , vK pξq ¯ , ´µ 4 , v0 ¯ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) and the pairs of MApξq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='33) are ˜ T λ ˜ |ξ|2 ´ |Aξ|2 λ2 ¸ ` dT dλ |Aξ|2 λ2 , Aξ ¸ , ˆT λ |ξ|2 , URπ{2ξ ˙ , ˆT λ |ξ|2 , v0 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) Since pFθGα,Aq pξq and MApξq are symmetric positive definite (s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ), LA pξq is diagonalizable and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we have some estimates of LA and its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given A P DAσ1,σ2, LA and its derivatives satisfy (i) σ2 σ2 1 |ξ|´1 ď µ ď σ1 σ2 2 |ξ|´1 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) zm |ξ|2 ď ∥MApξq∥ ď zp0q M |ξ|2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) σ2zm 4σ2 1 |ξ| ď ∥LApξq∥ ď σ1zp0q M 2σ2 2 |ξ| , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) where zm “ minσ1ďλď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2σ1 ´ T λ , ` T λ ` dT dλ ˘ σ2 2 λ2 ¯ , zp0q M “ maxσ1ďλď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2σ1 ` T λ ` dT dλ ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (ii) ���� BLA Bξk ���� ď C1 σ2 1 σ3 2 zp0q M , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) ���� B2LA BξkBξl ���� ď C1 σ3 1 σ4 2 zp0q M |ξ|´1 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) where C1 is a constant that does not depend on α or A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (iii) B Bξj ∆ξLApξq, p∆ξq2 LApξq and B Bξj p∆ξq2 LApξq may be written as B Bξj ∆ξLApξq “ 1 |ξ|2 Φp3q A,jpˆξq, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) p∆ξq2 LApξq “ 1 |ξ|3 Φp4q A pˆξq, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) B Bξj p∆ξq2 LApξq “ 1 |ξ|4 Φp5q A,jpˆξq, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) where Φp3q A,j, Φp4q A , Φp5q A,j are bounded on ���ˆξ ��� “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (i) We first note the following inequalities: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) σ2 ď ∥B∥ ď σ1, 1 σ1 ď ��B´1�� “ ∥U∥ ď 1 σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We thus have: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) σ2 2 ď detpBq “ ∥B∥ ��B´1��´1 ď σ2 1, 44 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN where we used the fact that B is a 2 ˆ 2 symmetric positive definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' From (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13), we immediately have: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15) |Uξ| “ ��B´1ξ �� ě 1 σ1 |ξ| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, σ2 σ2 1 |ξ|´1 ď µ “ 1 detpBq |B´1ξ| ď σ1 σ2 2 |ξ|´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, through (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4), one of the eigenvalues of MA is bounded by ˆT λ ` dT dλ ˙ σ2 2 λ2 |ξ|2 ď T λ ˜ |ξ|2 ´ |Aξ|2 λ2 ¸ ` dT dλ |Aξ|2 λ2 ď ˆT λ ` dT dλ ˙ |ξ|2 , so we may obtain (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, since LA is diagonalizable and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' with LA “ FθGα,AMA, the eigenvalues of LA are between µ 4 zm |ξ|2 and µ 2 zp0q M |ξ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, we get the bound (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (ii) We now turn to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Note that: B Bξk ˆ 1 |Uξ| ˙ “ ´ ` U TUξ ˘ k |Uξ|3 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16) B Bξk B Bξl ˆ 1 |Uξ| ˙ “ ´ ` U TU ˘ k,l |Uξ|3 ` 3 ` U TUξ ˘ k ` U TUξ ˘ l |Uξ|5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='17) Likewise, we have: B Bξk ˆpUξqj |Uξ| ˙ “ Ujk |Uξ| ´ pUξqj ` U TUξ ˘ k |Uξ|3 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18) B Bξk B Bξl ˆpUξqj |Uξ| ˙ “ ´Ujk ` U TUξ ˘ l |Uξ|3 ´ Ujl ` U TUξ ˘ k |Uξ|3 ´ pUξqj ` U TU ˘ k,l |Uξ|3 ` 3pUξqj ` U TUξ ˘ k ` U TUξ ˘ l |Uξ|5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='19) The above relations, together with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13), show that: ���� B Bξk ˆ 1 |Uξ| ˙���� ď σ2 1 σ2 1 |ξ|2 , ���� B Bξk B Bξl ˆ 1 |Uξ| ˙���� ď 4σ3 1 σ2 2 1 |ξ|3 , ���� B Bξk ˆpUξqj |Uξ| ˙���� ď 2σ1 σ2 1 |ξ|, ���� B Bξk B Bξl ˆpUξqj |Uξ| ˙���� ď 6σ2 1 σ2 2 1 |ξ|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, we obtain ���� BFθGα,A Bξk ���� ď C σ2 1 σ3 2 |ξ|´2 , ���� B2FθGα,A BξkBξl ���� ď C σ3 1 σ4 2 |ξ|´3 , Next, set A “ rA1A2s, since ���� B Bξk Aξ ���� “ |Ak| ď λ, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 45 we have ���� B Bξk pAξ b Aξq ���� ď Cσ1λ |ξ| , ���� B Bξk B Bξl pAξ b Aξq ���� ď Cλ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, ���� B Bξk MA ���� ď C ˆT λ ` dT dλ ˙ |ξ| , ���� B Bξk B Bξl MA ���� ď C ˆT λ ` dT dλ ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The desired bound (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) now follows easily by combining the above estimates and 1 ď σ1 σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (iii) By lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1, we may obtain B Bξj ∆ξLApξq “ ÿ i“1,2 B Bξjii LApξq “ ÿ i“1,2 1 |ξ|2 Pjii ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since σ2 ď ���Uˆξ ��� ď σ1 and Pjii ´ ˆξ1, ˆξ2 ¯ is a matrix of polynomials on the domain ���ˆξ ��� “ 1, Pjiipˆξ1,ˆξ2q |Uˆξ| 9 is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, we have Φp3q A,jpˆξq “ ÿ i“1,2 Pjii ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 9 , B Bξj ∆ξLApξq “ 1 |ξ|2 Φp3q A,jpˆξq, p∆ξq2 LApξq “ 1 |ξ|3 Φp4q A pˆξq, B Bξj p∆ξq2 LApξq “ 1 |ξ|4 Φp5q A,jpˆξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Similarly, Φp4q A pˆξq “ ÿ i,k“1,2 Pkkii ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 11 , Φp5q A,jpˆξq “ ÿ i,k“1,2 Pjkkii ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 13 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ 46 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Some estimates for z ` Lα,A and pz ` Lα,Aq´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since LA pξq is p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' and diagonalizable, P ´1 pξq LA pξq P pξq “ D pξq where D is a positive diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, P ´1 pξq pz ` LA pξqq P pξq “ z ` D pξq , P ´1 pξq pz ` LA pξqq´1 P pξq “ pz ` D pξqq´1 , and the eigenvalues of pz ` LA pξqq´1 are on a curve # 1 z ` a ˇˇˇ a P ”σ1zp0q M 2σ2 2 |ξ| , σ2zm 4σ2 1 |ξ| ı+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let λ ą 0 and z P Sω,δ with ω ą 0, β :“ π´argpz ` λ ´ ωq ą δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we obtain the following inequality |z ` λ|2 “λ2 ` |z|2 ´ 2 |z| λ cos β ěλ2 ` |z|2 ´ 2 |z| λ cos δ “ cos δ pλ ´ |z|q2 ` p1 ´ cos δq ´ λ2 ` |z|2¯ ě p1 ´ cos δq ´ λ2 ` |z|2¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Now, we estimate ���Bα ξ pz ` LA pξqq´1��� with |α| ď 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given Sω,δ with ω ą 0 and A P DAσ1,σ2, we have the following estimates for all z P Sω,δ, 1 |z| ` σ1zp0q M 2σ2 2 |ξ| ď ���pz ` LA pξqq´1��� ď 2 b p1 ´ cos δq ` p σ2zm 4σ2 1 |ξ|q2 ` |z|2˘, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20) ���� B Bξk pz ` LA pξqq´1 ���� ď C1 σ2 1 σ3 2 zp0q M 4 p1 ´ cos δq ` p σ2zm 4σ2 1 |ξ|q2 ` |z|2 ˘, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21) and ���� B2 BξlBξk pz ` LA pξqq´1 ���� ď C2 1 σ4 1 σ6 2 zp0q M 2 16 ˆ p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙˙ 3 2 ` C1 σ3 1 σ4 2 zp0q M 4 p1 ´ cos δq |ξ| ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22) Moreover, there exists a constant Cδ,σ1,σ2,T depending on δ, σ1, σ2 and T s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' for all |α| ď 2, ���Bα ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2,T |z| |ξ|´|α| , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='23) and ���Bα ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2,T |z|2 |ξ|1´|α| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24) WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 47 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since the eigenvalues of pz ` LA pξqq´1 are between ´ z ` σ1zp0q M 2σ2 2 |ξ| ¯´1 and ´ z ` σ2zm 4σ2 1 |ξ| ¯´1 , it follows that 1 |z| ` σ1zp0q M 2σ2 2 |ξ| ď ���pz ` LA pξqq´1��� ď 2 d p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, B Bξk pz ` LA pξqq´1 “ ´ pz ` LA pξqq´1 B Bξk LA pξq pz ` LA pξqq´1 , so by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8), ���� B Bξk pz ` LA pξqq´1 ���� ď C1 σ2 1 σ3 2 zp0q M 4 p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, B2 BξlBξk pz ` LA pξqq´1 “ pz ` LA pξqq´1 B Bξl LA pξq pz ` LA pξqq´1 B Bξk LA pξq pz ` LA pξqq´1 ` pz ` LA pξqq´1 B Bξl LA pξq pz ` LA pξqq´1 B Bξk LA pξq pz ` LA pξqq´1 ´ pz ` LA pξqq´1 B2 BξlBξk LA pξq pz ` LA pξqq´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, ���� B2 BξlBξk pz ` LA pξqq´1 ���� ď C2 1 σ4 1 σ6 2 zp0q M 2 16 ˆ p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙˙ 3 2 ` C1 σ3 1 σ4 2 zp0q M 4 p1 ´ cos δq |ξ| ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' From the inequalities 1 dˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙ ď 1 |z| and 1 dˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙ ď 1 σ2zm 4σ2 1 |ξ|, we obtain ���Bα ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2 |z| |ξ|´|α| , ���Bα ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2 |z|2 |ξ|1´|α| for all |α| ď 2, where the constant Cδ,σ1,σ2,T only depends on δ, σ1, σ2 and T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Next, let us consider ���Bα ξ |ξ| pz ` LA pξqq´1��� with |α| ď 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 48 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given Sω,δ with ω ą 0 and A P DAσ1,σ2, the following estimates hold for all z P Sω,δ ���|ξ| pz ` LA pξqq´1��� ď 2 |ξ| d p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='25) ���� B Bξk ´ |ξ| pz ` LA pξqq´1¯���� ď 2 d p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙ ` C1 σ2 1 σ3 2 zp0q M 4 |ξ| p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='26) ���� B2 BξlBξk ´ |ξ| pz ` LA pξqq´1¯���� ď 4 |ξ| d p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙ ` C1 σ3 1 σ4 2 zp0q M 12 p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙ ` C2 1 σ4 1 σ6 2 zp0q M 2 16 |ξ| ˆ p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙˙ 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='27) Moreover, there exists a constant Cδ,σ1,σ2,T depending on δ, σ1, σ2 and T s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' for all |α| ď 2, ���Bα ξ |ξ| pz ` LA pξqq´1��� ď Cδ,σ1,σ2,T |ξ|´|α| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='28) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20), it is easy to obtain ���|ξ| pz ` LA pξqq´1��� ď 2 |ξ| d p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, B Bξk ´ |ξ| pz ` LA pξqq´1¯ “B |ξ| Bξk pz ` LA pξqq´1 ` |ξ| B Bξk pz ` LA pξqq´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 49 Therefore, with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21), ���� B Bξk ´ |ξ| pz ` LA pξqq´1¯���� ď 2 d p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙ ` C1 σ2 1 σ3 2 zp0q M 4 |ξ| p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, B2 BξlBξk ´ |ξ| pz ` LA pξqq´1¯ “ B2 BξlBξk |ξ| pz ` LA pξqq´1 ` B Bξk |ξ| B Bξl pz ` LA pξqq´1 ` B Bξl |ξ| B Bξk pz ` LA pξqq´1 ` |ξ| B2 BξlBξk pz ` LA pξqq´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, ���� B2 BξlBξk ´ |ξ| pz ` LA pξqq´1¯���� ď 4 |ξ| d p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙ ` C1 σ3 1 σ4 2 zp0q M 12 p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙ ` C2 1 σ4 1 σ6 2 zp0q M 2 16 |ξ| ˆ p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙˙ 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' With 1 dˆˆ σ2zm 4σ2 1 |ξ| ˙2 `|z|2 ˙ ď 1 σ2zm 4σ2 1 |ξ|, we obtain ���Bα ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2,T |ξ|´|α| for all |α| ď 2, where the constant Cδ,σ1,σ2,T only depends on δ, σ1, σ2 and T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Now, we may prove Lα,A is a sectorial operator and obtain the estimate of pz ´ Lα,Aq´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given a matrix A satisfying the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) and a constant K ą 0 , there exists Sω,δ with ω, δ ą 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' for all z P Sω,δ ���pz ´ Lα,Aq´1 Y ��� CγpR2q ď Cω,δ,σ1,σ2,T |z| ∥Y ∥CγpR2q , and ���pz ´ Lα,Aq´1 Y ��� C1,γpR2q ď Cω,δ,σ1,σ2,T ∥Y ∥CγpR2q , for all Y P CγpR2q X LppR2q with 1 ď p ď 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 50 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since Lα,AY pθq “ ´F´1LApξqFY , the Fourier multiplier of pz ´ Lα,Aq´1 is pz ` LAq´1 and of B Bθi pz ´ Lα,Aq´1 is ξi pz ` LAq´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' With (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='23) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='28), we obtain there exists Cω,δ,σ1,σ2,T s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' �pz ´ Lα,Aq´1 Y �CγpR2q ďCω,δ,σ1,σ2,T |z| �Y �CγpR2q, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='29) �pz ´ Lα,Aq´1 Y �C1,γpR2q ďCω,δ,σ1,σ2,T �Y �CγpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='30) Next, for ���pz ´ Lα,Aq´1 Y ��� C0pR2q, set ϕpξq to be a smooth and radial cutting func- tion with a compact support in B p1q and ϕpξq “ 1 in a neighborhood of ξ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, ���pz ´ Lα,Aq´1 Y ��� C0pR2q “ ���F´1 pz ` LAq´1 pξqFY ��� C0pR2q ď ���F´1 pz ` LAq´1 pξq p1 ´ ϕpξqq FY ��� C0pR2q ` ���F´1 pz ` LAq´1 pξqϕpξqFY ��� C0pR2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the first term, since 1´ϕpξq “ 0 in a neighborhood of ξ “ 0 and |1 ´ ϕpξq| ď 1, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3, we obtain ���F´1 pz ` LAq´1 pξq p1 ´ ϕpξqq FY ��� C0pR2q ďC�F´1 pz ` LAq´1 pξq p1 ´ ϕpξqq FY �CγpR2q ď Cω,δ,σ1,σ2,T |z| �Y �CγpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the second term, define the kernel K0 pθq :“ F´1 pz ` LAq´1 pξqϕpξq, so ���F´1 pz ` LAq´1 pξqϕpξqFY ��� C0pR2q “ ∥K0 ˚ Y ∥C0pR2q ď ∥K0∥L1pR2q ∥Y ∥C0pR2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we estimate ∥K0∥L1, by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2, we have ∥K0 pθq∥ ďCω,δ,σ1,σ2,T |z| 1 1 ` |θ|3 , so ∥K0∥L1 ď Cω,δ,σ1,σ2,T |z| ż R2 1 1 ` |θ|3 dθ ď Cω,δ,σ1,σ2,T |z| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, ���F´1 pz ` LAq´1 pξqϕpξqFY ��� C0pR2q ď Cω,δ,σ1,σ2,T |z| ∥Y ∥C0pR2q , so ���pz ´ Lα,Aq´1 Y ��� CγpR2q ď Cω,δ,σ1,σ2,T |z| ∥Y ∥CγpR2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 51 Similarly, for ��� B Bθi pz ´ Lα,Aq´1 Y ��� C0pR2q, we may use the above technique with the kernel K1,i K1,i pθq :“F´1ξi pz ` LAq´1 pξqϕpξq, ∥K1,j pθq∥ ďCω,δ,σ1,σ2,T 1 1 ` |θ|4 to obtain ���� B Bθi pz ´ Lα,Aq´1 Y ���� C0pR2q ď Cω,δ,σ1,σ2,T ∥Y ∥C0pR2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, ���pz ´ Lα,Aq´1 Y ��� C1,γpR2q ď Cω,δ,σ1,σ2,T ∥Y ∥CγpR2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given a matrix A in DAσ1,σ2, there exists Sω,δ with ω, δ ą 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' for all z P Sω,δ ∥pz ´ Lα,Aq Y ∥CγpR2q ě Cω,δ,σ1,σ2,T |z| ∥Y ∥CγpR2q , and ∥pz ´ Lα,Aq Y ∥CγpR2q ě Cω,δ,σ1,σ2,T ∥Y ∥C1,γpR2q , for all compactly supported Y P C1,γ ` R2˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given Y P C1,γ with a compact support, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5, W “ pz ´ Lα,Aq Y P Cγ ` R2˘ X L2 ` R2˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Through Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10, ∥W ∥CγpR2q ě Cp1q ω,δ,σ1,σ2,T |z| ���pz ´ Lα,Aq´1 W ��� CγpR2q “ Cp1q ω,δ,σ1,σ2,T |z| ∥Y ∥CγpR2q (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='31) and ∥W ∥CγpR2q ě Cp2q ω,δ,σ1,σ2,T �pz ´ Lα,Aq´1 W �C1,γpR2q “ Cp2q ω,δ,σ1,σ2,T ∥Y ∥C1,γpR2q (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='32) □ In each chart x Xn pθq and Y n pθq, A is ∇Xn p0q and ρnY n is supported in V4R (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lα,A will be used in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since Apξ “ lim hÑ0 Xnphpξq ´ Xnp0q h , it is clear that |X|˚ ď |X|˝,n ď lim inf ξÑ0 |Xn pξq ´ Xn p0q| |ξ| ď ���Apξ ��� , C ∥X∥C1,γpS2q ě ∥Xn∥C1pV4Rq ě lim sup ξÑ0 |Xn pξq ´ Xn p0q| |ξ| ě ���Apξ ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, we may set σ1 “ C ∥X∥C1,γpS2q , σ2 “ |X|˚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 52 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Next, set A0 “ ∇x Xn p0q, so A0 “ » – 2 0 0 2 0 0 fi fl , and L0,A0Y pθq “ ´ ż R2 B Bηk 1 4π |θ|W kdη, L0,A0 pξq “ |ξ| 8 I Set Lpσq A “ p1 ´ σq L0,A0 ` σL0,A, which will be used in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6 , then Lpσq A Y pθq “ ´ 1 8π ż R2 B Bηk ˆ2 p1 ´ σq |θ| ` σ |Aθ| ˙ W kdη, Lpσq A pξq “|ξ| 4 ˆ1 ´ σ 2 ` σ |ξ| det pBq |Uξ| ˙ We just have to adapt the above theorems and their proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, we obtain Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given X P C1,γpS2q, and our Stereoghraphic projection charts and the partition functions tx Xn, ρnu with the radius R, set σ1 “ C ∥X∥C1,γpS2q, σ2 “ |X|˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' There exists Sω,δ with ω, δ ą 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' in each chart x Xn, for Y P C1,γpS2q, we have the following inequalities: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='33) ∥pz ´ Lα,Aq ρnY n∥CγpR2q ě Cp1q ω,δ,σ1,σ2,T |z| ∥ρnY n∥CγpR2q , ∥pz ´ Lα,Aq ρnY n∥CγpR2q ě Cp2q ω,δ,σ1,σ2,T ∥ρnY n∥C1,γpR2q , ��� ´ z ´ Lpσq A ¯ ρnY n ��� CγpR2q ě Cp3q ω,δ,σ1,σ2,T |z| ∥ρnY n∥CγpR2q , ��� ´ z ´ Lpσq A ¯ ρnY n ��� CγpR2q ě Cp4q ω,δ,σ1,σ2,T ∥ρnY n∥C1,γpR2q , where A “ ∇Xn p0q, and σ, α P p0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Some estimates for etLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We have two ways of representing the semigroup e´tLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' One is by the Dunford integral etLA “ 1 2πi ż ω`γr,η etz pz ` LAq´1 dz, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='34) where r ą 0, δ ă η ă π 2 and the curve γr,η “ tz P C : |argz| “ π ´ η, |z| ě ru X tz P C : |argz| ď π ´ η, |z| “ ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The other is through Fourier transform, etLAf pθq :“ F´1 ” e´tLApξqFrfs pξq ı pθq “ ż R2 KA pt, θ ´ ηq f pηq dη, where KA pt, θq “ F´1 “ e´tLApξq‰ pθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given 0 ď t0 ď t ď T and 0 ď β´α ă 1 2, we have the following estimates: }ept´t0qLAfpt0q}CβpR2q ď C pt ´ t0qβ´α }fpt0q}CαpR2q, }ept´t0qLAfpt0q}L2pt0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='CβpR2qq ď C}fpt0q}CαpR2q, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 53 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='34), Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10 and 0 ď t ´ t0 ď T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' we have ���ept´t0qLAfpt0q ��� CαpR2q “ ����� 1 2πi ż ω`γr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='η ept´t0qz pz ` LAq´1 fpt0qdz ����� CαpR2q ď Cω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='T ∥fpt0q∥CαpR2q ż ω`γr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='η ���ept´t0qz��� 1 |z|d |z| ď Cω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='T ∥fpt0q∥CαpR2q ż pt´t0qpω`γr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ηq |ez| 1 |z|d |z| ď Cω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='T ∥fpt0q∥CαpR2q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' and ���ept´t0qLAfpt0q ��� C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='αpR2q ď Cω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='T ∥fpt0q∥CαpR2q ż ω`γr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='η ���ept´t0qz��� d |z| ď Cω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='T t ´ t0 ∥fpt0q∥CαpR2q ż pt´t0qpω`γr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ηq |ez| d |z| ď Cω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='T t ´ t0 ∥fpt0q∥CαpR2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, by interpolation theorem, we obtain ���ept´t0qLAfpt0q ��� CβpR2q ď Cω,δ,σ1,σ2,T ,T,β´α pt ´ t0qβ´α ∥fpt0q∥CαpR2q , so }ept´t0qLAfpt0q}L2pt0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='CβpR2qq ď Cω,δ,σ1,σ2,T ,T,β´α}fpt0q}CαpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given 0 ď t0 ď t ď T and 0 ď α ă 1, we have the following estimates: ���� ż t t0 ept´sqLAfpsqds ���� C1,αpR2q ď C sup t0ďsďt }fpsq}CαpR2q, ���� ż t t0 ept´sqLAfpsqds ���� L2pt0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C1,αpR2qq ď C}f}L2pt0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='CαpR2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' First, define u as upt, t0, θq :“ urfs pθq“ ż t t0 ept´sqLAfps, θqds “ ż t t0 ż R2KApt´s, θ´ηqfps, ηqdηds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since tLA pξq “ LA ptξq, KApt ´ s, x ´ yq “ 1 pt ´ sq2 KA ˆ 1, θ ´ η t ´ s ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 54 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN By (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='48) in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3, |upt, t0, θq| “ ����� ż t t0 ż R2 1 pt ´ sq2 KA ˆ 1, θ ´ η t ´ s ˙ fps, ηqdηds ����� ď ż t t0 ż R2 1 pt ´ sq2 C 1 ` ��� θ´η t´s ��� 3 |fps, ηq| dηds ďC ∥f∥C0 ż t t0 ż R2 1 1 ` |θ ´ η|3 dηds ď C ∥f∥C0 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, we assume f ps, θq “ 0 when s ă t0, so Bu Bθi “ ż t t0 ż R2 1 pt ´ sq3 BKA Bθi ˆ 1, θ ´ η t ´ s ˙ fps, ηqdηds “ ż t ´8 ż R2 1 pt ´ sq3 BKA Bθi ˆ 1, θ ´ η t ´ s ˙ fps, ηqdηds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Through (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='49), ż R2 1 pt ´ sq3 BKA Bθi ˆ 1, θ ´ η t ´ s ˙ fps, ηqdη “ ż R2 1 pt ´ sq3 BKA Bθi ˆ 1, θ ´ η t ´ s ˙ pfps, ηq ´ f ps, θqq dη, and ����� ż R2 1 pt ´ sq3 BKA Bθi ˆ 1, θ ´ η t ´ s ˙ pfps, ηq ´ f ps, θqq dη ����� ď 1 pt ´ sq3 ż R2 |fps, ηq ´ f ps, θq| 1 ` ��� θ´η t´s ��� 4 dη ď C�f ps, ¨q�Cα 1 pt ´ sq3 ż R2 |θ ´ η|α 1 ` ��� θ´η t´s ��� 4 dη “ C�f ps, ¨q�Cα 1 pt ´ sq1´α ż R2 |θ ´ η|α 1 ` |θ ´ η|4 dη “ C�f ps, ¨q�Cα 1 pt ´ sq1´α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5, for all m ď t, we obtain ����� ż t m ż R2 1 pt ´ sq3 BKA Bθi ˆ 1, θ ´ η t ´ s ˙ fps, ηqdηds ����� ďC ż t m �f ps, ¨q�Cα 1 pt ´ sq1´α ds ď C pt ´ mqα Ml r�f ps, ¨q�Cαs ptq , so set m “ 0, ���� Bu Bθi pt, t0, θq ���� ď CT αMl r�f ps, ¨q�Cαs ptq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 55 For � Bu Bθi �Cα, we first compute B2 BθiBθj ż M ´8 ż R2KApt ´ s, θ ´ ηqfps, ηqdηds “ ż M 8 ż R2 1 pt ´ sq4 B2KA BθiBj ˆ 1, θ ´ η t ´ s ˙ fps, ηqdηds, where M ă t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='50) and Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5, we have ����� ż R2 1 pt´sq4 B2KA BθiBj ´ 1, θ ´ η t´s ¯ fps, ηqdη ����� ď C�f ps, ¨q�Cα 1 pt´sq2´α ż R2 |θ´η|α 1`|θ´η|4 dη ď C�f ps, ¨q�Cα 1 pt ´ sq2´α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' and ����� B2 BθiBθj ż M ´8 ż R2 KApt ´ s, θ ´ ηqfps, ηqdηds ����� ď C pt ´ Mqα´1 Ml r�f ps, ¨q�Cαs ptq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' When |θ ´ η| “ 1, we set a cutting function φpsq s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' φpsq “ 1 on r´8, ´2s and φpsq “ 0 on r´1, 8s, and define φptqpsq “ φps ´ tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We obtain ���� Bu Bθi pt, t0, θq ´ Bu Bηi pt, t0, ηq ���� “ ���� Bu Bθi rfspθq ´ Bu Bηi rfspηq ���� ď ���� Bu Bθi rφptqfspθq ´ Bu Bηi rφptqfspηq ����` ���� Bu Bθi rp1´φptqqfspθq ����` ���� Bu Bηi rp1´φptqqfspηq ���� ď Cpt´Mqα´1Ml “ �φptqfps, ¨q�Cα‰ ptq` Cpt´mqαMl “ �p1´φptqqfps, ¨q�Cα‰ ptq, where M “ t ´ 1 and m “ t ´ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, since 0 ď φ ď 1 and �f ps, ¨q�Cα ě 0, ���� Bu Bθi pt, t0, θq ´ Bu Bηi pt, t0, ηq ���� ď CMl r�f ps, ¨q�Cαs ptq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' When ρ “ |θ ´ η| ‰ 1, we define uρ pt, t0, θq “ 1 ρu pρt, ρt0, ρθq , f ρ pt, θq “ f pρt, ρθq and ¯θ “ θ ρ , ¯η “ η ρ , ¯t “ t ρ, ¯t0 “ t0 ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We have Buρ Bt pt, θq “LAuρ pt, θq ` f ρ pt, θq , Buρ Bθi pt, θq “ Bu Bθi pρt, ρθq , so ˇˇˇ Bu Bθi pt, t0, θq´ Bu Bηi pt, t0, ηq ˇˇˇ“ ˇˇˇBuρ B¯θi p¯t, ¯t0, ¯θq´ Buρ B¯ηi p¯t, ¯t0, ¯ηq ˇˇˇďCMlr�f ρ p¯s, ¨q�Cαsptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since �f ρ�Cα p¯sq “ ρα�f�Cα pρ¯sq, Ml “ �f ρ p¯s, ¨q�Cα‰ p¯tq “ sup ¯rą0 1 ¯r ż ¯t ¯t´¯r �f ρ�Cα p¯sq d¯s “ ρα sup ¯rą0 1 ¯r ż ¯t ¯t´¯r �f�Cα pρ¯sq d¯s “ρα sup rą0 1 r ż t t´r �f�Cα psq ds “ ραMl r�f ps, ¨q�Cαs ptq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 56 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Hence, ���� Bu Bθi pt, t0, θq ´ Bu Bηi pt, t0, ηq ���� ď CMl r�f ps, ¨q�Cαs ptq |θ ´ η|α , and by the Hardy-Littlewood maximal function theorem, for all 1 ď p ď 8 ���∥u∥CγpR2q ��� Lppt0,T q ď ���C ∥f∥C0pR2q ` CMl “ �f ps, ¨q�CαpR2q ‰ ptq ��� Lppt0,T q ďC ���∥f∥C0pR2q ��� Lppt0,T q ` C ��Ml “ �f ps, ¨q�CαpR2q ‰ ptq �� Lppt0,T q ďC ���∥f∥C0pR2q ��� Lppt0,T q ` C ���f pt, ¨q�CαpR2q �� Lppt0,T q ďC ���∥f∥CγpR2q ��� Lppt0,T q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Local well-posedness We write the Peskin problem as an evolution equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) BX Bt “ FpXq, t ą 0, Xp0q “ X0, where FpXq is given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will make use of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 in [35]: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let E1 Ă E0 Ă E be Banach spaces and let 0 ă σ ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given T ą 0, open set O1 Ă E1 and a function F : r0, T s ˆ O1 ÞÑ E0, pt, uq ÞÑ Fpt, uq such that F and Fu are continuous in r0, T s ˆ O1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If for every p¯t, ¯uq P r0, T s ˆ O1 we have Fup¯t, ¯uq : E1 ÞÑ E0 is the part of a sectorial operator S : DpSq Ă E ÞÑ E with DSpσq » E0 and DSpσ ` 1q » E1, then for every ¯t P r0, ts and ¯u P O1 there are δ ą 0, r ą 0 such that if t0 P r0, T q, |t0 ´ ¯t| ď δ, and }u0 ´ ¯u} ď r then the problem v1ptq “ Fpt, vptqq, t0 ď t ď t0 ` δ, vpt0q “ u0, has a unique solution v P Cprt0, t0 ` δs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' E1q X C1prt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t0 ` δs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' E0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, our main result is the following Theorem: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Consider the 3D Peskin problem (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) with initial data satisfying X0 P h1,γpS2q, |X0|˚ ą 0, and T P C3 such that T ą 0, dT {dλ ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, there exists some time T ą 0 such that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) has a unique solution X, X P Cpr0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' h1,γpS2qq X C1pr0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' hγpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let Om “ tY P h1,γpS2q : |Y |˚ ě m ą 0u, E1 “ h1,γpS2q, E0 “ hγpS2q, and E “ hαpS2q, with 0 ă α ă γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Define the operator S as the linearization of F (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) around X0: SpX0qY :“ BXFpX0qY “ d dεFpX0 ` εY q|ε“0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since X0 P Om is arbitrary, we can study the Gˆateaux derivative of F at any X P Om, which is given by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) SpXqY “ S1pXqY ` S2pXqY , WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 57 with S1pXqY “ ´ ż S2 ∇S2GpXppxq ´ Xppyqq¨ ˆ d dε ´ T p|∇S2pXppyq ` εY ppyqq|qp∇S2pX ` εY qqppyq ¯ |ε“0dpy, S2pXqY “ ´ ż S2∇S2 d dε ´ GpXppxq´Xppyq`εpY ppxq ´ Y ppyqqq ¯ˇˇˇ ε“0¨ ˆ T p|∇S2Xppyq|q∇S2Xppyqdpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It remains to check that the hypothesis of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 are satisfied, which follow from Propositions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If m ą 0 and γ P p0, 1q, T P C2, then F (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) is a continuous map from Om Ă h1,γpS2q to hγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given that FpXq “ NpXqpT p|∇S2X|q∇S2Xq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5), we apply Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5 to obtain that }FpXq}CγpS2q ď C 1 |X|˚ ´ 1 ` ´}∇S2X}C0pS2q |X˚| ¯2¯ }T p|∇S2Xq|q∇S2Xq}CγpS2q, hence recalling the expression for T (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4), the bound above yields that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) }FpXq}CγpS2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2X}CγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We have thus proved that F maps C1,γpS2q to CγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We need to show that it also maps h1,γpS2q to hγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Having the estimate (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3), it suffices to show that if X P h1,γpS2q, then FpXq P hγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since hk,γpS2q is the completion of Ck,γpS2q in any Ck,αpS2q with 0 ă γ ă α ă 1, k ě 0, let X P h1,γpS2q, and tXmum a sequence Xm P C1,αpS2q, α ą γ, such that Xm Ñ X in C1,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It is clear that the previous estimate (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) also holds replacing γ by α, thus FpXmq P Cα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We conclude that FpXq P hγpS2q by showing that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) }FpXmq ´ FpXq}CγpS2q ď C}Xm ´ X}C1,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The estimate will follow from the previous ones by writing FpXmq ´ FpXq as follows: (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) FpXmqppxq ´ FpXqppxq “ ∆1ppxq ` ∆2ppxq, with ∆1ppxq “ ´ ż S2∇S2GpXmppxq´Xmppyqq¨ ˆ ` T p∇S2Xmppyqq∇S2Xmppyq´T p∇S2Xppyqq∇S2Xppyq ˘ dpy, ∆2ppxq “ ż S2∇S2 ´ GpXmppxq´Xmppyqq´GpXppxq´Xppyqq ¯ ¨ ˆ T p∇S2Xppyqq∇S2Xppyqdpy, 58 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN where both terms have kernels given by a derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The first term has thus already been treated, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) }∆1}CγpS2q ď Cp}∇S2Xm}C0pS2q, |Xm|˚q}T p∇S2Xmq∇S2Xm´T p∇S2Xq∇S2X}CγpS2q ďCp}∇S2Xm}C0pS2q, |Xm|˚, }∇S2X}C0pS2q, |X|˚, }T }C2q ´ }∇S2pXm´Xq}CγpS2q ` p}∇S2X}CγpS2q ` }∇S2Xm}CγpS2qq}∇S2pXm ´ Xq}C0pS2q ¯ , while the second one can be estimated in a similar manner by noticing that one can always extract Xm ´ X from the difference of kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Consider for example the kernel q1 k,l (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7), q1 k,lppx, pyq “ ´ 1 8π δ pyXippxq |δ pyXppxq|3 ∇S2Xippyqδk,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We can write 1 8π δ pyXm i ppxq |δ pyXmppxq|3 ∇S2Xm i ppyqδk,l ´ 1 8π δ pyXippxq |δ pyXppxq|3 ∇S2Xippyqδk,l “ 1 8π δ pypXm i ´ Xiqppxq∇S2Xm i ppyq |δ pyXmppxq|3 ` 1 8π δ pyXippxq∇S2pXm i ´ Xiqppyq |δ pyXmppxq|3 ` 1 8π δ pyXippxq∇S2Xippyq ´ 1 |δ pyXmppxq|3 ´ 1 |δ pyXppxq|3 ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, it holds that }∆2}CγpS2q ď Cp}∇S2X}C0pS2q, |X|˚, }∇S2Xm}C0pS2q, |Xm|˚, }T }C1q ˆ }∇S2X}CγpS2q}∇S2pXm ´ Xq}C0pS2q, which together with (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) proves (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If m ą 0 and γ P p0, 1q, T P C3, then the Gˆateaux derivative of F at any X P Om Ă h1,γpS2q (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) is continuous and maps h1,γpS2q to hγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The first term S1pXqY in the Gˆateaux derivative of F (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) is given in terms of the operator NpXq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) S1pXqY ppxq “ NpXqpTSp∇S2Xq∇S2Y qppxq, with TS given by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) TSp∇S2Xq“ T p|∇S2X|q |∇S2X| ` ´ T 1p|∇S2X|q´ T p|∇S2X|q |∇S2X| ¯∇S2X b ∇S2X |∇S2X|2 , and, in index notation, pTSp∇S2Xq∇S2Y ql,i “ T p|∇S2X|q |∇S2X| p∇S2Y ql,i ` ´ T 1p|∇S2X|q´ T p|∇S2X|q |∇S2X| ¯p∇S2Xql,ip∇S2Xqq,m |∇S2X|2 p∇S2Y qq,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 59 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5 then gives that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) }S1pXqY }CγpS2q ď Cp|X|˚, }∇S2X}C0pS2qq}TSp∇S2Xq∇S2Y }CγpSq ď Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2Y }CγpSq ` Cp|X|˚, }∇S2X}C0pS2q, }T }C2q}∇S2X}CγpS2q}∇S2Y }C0pSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We proceed with S2pXqY , S2pXqY “ S2,1pXqY ` S2,2pXqY , pS2,jpXqY qkppxq “ ż S2 Qj k,lppx, pyq ¨ pT p∇S2Xq∇S2Xlppyq ´ Clqdpy, where we define the kernels Qj k,lppx, pyq “ ∇S2 d dε ´ GjpXppxq ´ Xppyq ` εpY ppxq ´ Y ppyqqq ¯ˇˇˇ ε“0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Taking the derivatives we see that Qj k,lppx, pyq “ ∇S2 ´ B Bxi GjpXppxq ´ XppyqqpYippxq ´ Yippyqq ¯ “ ´ B Bxl B Bxi GjpXppxq ´ Xppyqq∇S2XlppyqpYippxq ´ Yippyqq ´ B Bxi GjpXppxq ´ Xppyqq∇S2Yippyq, hence we have the following bound, similarly as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) |Qj k,lppx, pyq| ď C ´ |∇S2Y ppyq| |∆ pyXppxq|2 ` |∇S2Xppyq||∆ pyY ppxq| |∆ pyXppxq|3 ¯ ď C }∇S2Y }C0pS2q |X|2˚ ´ 1 ` }∇S2X}C0pS2q |X|˚ ¯ 1 |px ´ py|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, |S2,jpXqY ppxq|ďC }T p∇S2Xq∇S2X}CγpS2q |X|2˚ ´ 1` }∇S2X}C0pS2q |X|˚ ¯ }∇S2Y }C0pS2q, hence |S2,jpXqY ppxq| ď Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2X}CγpS2q}∇S2Y }C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given that the kernels Qj k,l are also a derivative, the estimate of the H¨older semi- norm follows the same steps as in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In fact, performing the splitting as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9), we find that rS2,jpXqY sCγpS2q ď C }T p∇S2Xq∇S2X}CγpS2q |X|2˚ ´ 1 ` ´}∇S2X}C0pS2q |X˚| ¯2¯ }∇S2Y }C0pS2q, thus (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) }S2pXqY }CγpS2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2X}CγpS2q}∇S2Y }C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 60 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Together with (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9), this shows that SpXq maps C1,γpS2q to CγpS2q, }SpXqY }CγpS2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C2q}∇S2X}CγpS2q}∇S2Y }C0pS2q ` Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2Y }CγpSq ď Cp|X|˚ , }∇S2X}CγpS2q, }T }C2q ∥∇S2Y ∥CγpS2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) We are left to show that SpXq also maps h1,γpS2q to hγpS2q and that it is continuous with respect to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We follow the lines below (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It suffices to show that if Y P h1,γpS2q, then SpXqY P hγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let Y P h1,γpS2q, and tY mum a sequence Y m P C1,αpS2q, α ą γ, such that Y m Ñ Y in C1,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since SpXqY m P CαpS2q, we conclude that SpXqY P hγpS2q by showing that }SpXqY m ´ SpXqY }CγpS2q ď C}Y m ´ Y }C1,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' But since we are dealing with a linear operator, the estimate is trivially satisfied from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' That the Gˆateaux derivative is continuous in X follows along the lines below (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In fact, ` SpX1q ´ SpX2q ˘ Y “ pS1pX1q ´ S1pX2qqY ` pS2pX1q ´ S2pX2qqY , and we decompose each Sj as in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, it is not hard to see that the following bound holds }pSpX2q ´ SpX1qqY }CγpS2q ď Cp}∇S2X1}C0pS2q, |X1|˚, }∇S2X2}C0pS2q, |X2|˚, }T }C3q ˆ ´ }∇S2Y }CγpS2q}∇S2pX1 ´ X2q}CγpS2q ` }∇S2Y }C0pS2q}∇S2pX1 ´ X2q}C0pS2qp}∇S2X1}CγpS2q ` }∇S2X2}CγpS2qq ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Consider the linear operator SpXq : C1,γpS2q Ñ CγpS2q defined in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) with X P C1,γpS2q, T P C2, T ą 0, dT {dλ ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, there exists a sector such that for all z in the sector }z ´ SpXqY }CγpS2q ě Cp}Y }C1,γpS2q ` |z|}Y }CγpS2qq, where the constant C depends only on the sector, γ, the norms }X}C1,γpS2q and }T }C2, and the arc-chord condition |X|˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' From (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2), we have (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) }pz ´ SpXqqY }CγpS2q ě }pz ´ S1pXqqY }CγpS2q ´ }S2pXqY }CγpS2q, and using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) we obtain that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) }pz ´ SpXqqY }CγpS2q ě }pz ´ S1pXqqY }CγpS2q ´ C ∥∇S2Y ∥C0pS2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We use the notation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we can write S1pXq (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) as (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) S1pXqY ppxq “ ´ ż S2 ∇S2GpXppxq ´ Xppyqq ¨ pTSp∇S2Xppyqq∇S2Y ppyq ´ Cqdpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 61 Next, we introduce the partition of unity ρn (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) to write S1pXqY ppxq “ ÿ n S1pXq pρnY q ppxq “´ ÿ n ż S2∇S2GpXppxq´Xppyqq¨pTSp∇S2Xppyqq∇S2pρnY qppyq´Cnqdpy, where now we will choose Cn “ 0 or Cn “ TSp∇S2Xppxqq∇S2pρnY qppxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will extensively use that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15) supp ρn Ă Bpxn,2R X S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We notice that ρnppxqzY ppxq ´ S1pXqpρnY qppxq “ ρnppxq ´ zY ppxq ´ S1pXqY ppxq ¯ ` ´ ρnppxqS1pXqY ´ S1pXqpρnY qppxq ¯ , hence 2}ρn}CγpS2q}pz ´ S1pXqqY }CγpS2q ě }ρn ´ zY ´ S1pXqY ¯ }CγpS2q ě }zρnY ´ S1pXqpρnY q}CγpS2q ´ }ρnS1pXqY ´ S1pXqpρnY q}CγpS2q, and summing in n we obtain (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16) }pz ´ S1pXqqY }CγpS2q ě C ÿ n p}I1 n}CγpS2q ´ }I2 n}CγpS2qq, where (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='17) I1 n “ zρnY ´ S1pXqpρnY q, I2 n “ ρnS1pXqY ´ S1pXqpρnY q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Recalling that S1pXq is given in terms of NpXq (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7), we split I2 n further: I2 n “ rρn, NpXqspTSp∇S2Xq∇S2Y q ` NpXq ` Tsp∇S2XqY b ∇S2ρn ˘ , and by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9, }I2 n}CγpS2q ď Cp|X|˚, }∇S2X}C0pS2qq ´ }∇S2ρn}C0pS2q}TSp∇S2Xq∇S2Y }C0pS2q ` }Tsp∇S2XqY b ∇S2ρn}CγpS2q ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, }I2 n}CγpS2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C2q}∇S2ρn}CγpS2q ˆ p}∇S2X}CγpS2q}Y }CγpS2q`}∇S2Y }C0pS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We proceed to deal with the term I1 n (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We introduce the cutoff (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) so that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18) }I1 n}CγpS2q ě }zρnY ´pρnS1pXqpρnY q}CγpS2q´ }p1 ´ pρnqS1pXqpρnY q}CγpS2q “ }I1,1 n }CγpS2q ´ }I1,2 n }CγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The last term will be smoother because the integral is not singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In fact, recalling again the expression of S1pXq in terms of NpXq (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7), we use Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7 to obtain }I1,2 n }CγpS2q ď CpR, |X|˚, }∇S2X}C0pS2qq}TSp∇S2Xq∇S2pρnY q}C0pS2q ď CpR, |X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2pρnY q}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 62 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Although the constant in the bound above (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16) becomes large for R small, it will suffice since it is lower order in terms of regularity for Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, we proceed to estimate I1,1 n (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We can decompose further by intro- ducing the frozen-coefficient linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We denote by x Xn the stereographic projection centered at pxn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pxn “ x Xnp0q, and Xnpθq “ Xpx Xnpθqq (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Recalling (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='17), we have (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='19) }I1,1 n }CγpS2q ě C}zρnY n ´ pρnNpXqpTSp∇S2Xq∇S2pρnY qqn}CγpR2q, and I1,1 n pθq “ zρnpθqY npθq ´ pρnpθqNpXqpTSp∇S2Xq∇S2pρnY qqnpθq “ J3 ` J4 ` J5 ` J6, with J3 “ zρnpθqY npθq ´ LApρnY nqpθq, J4 “ pρnpθqrMpAq ´ NpXqspTSp∇S2Xq∇S2pρnY qqnpθq “ pρnpθqrMpAq ´ Mp∇Xnq ´ RnpXnqspTSp∇S2Xq∇S2pρnY qqnpθq, J5 “ pρnpθq ´ LApρnY nqpθq ´ MpAqpTSp∇S2Xqq∇S2pρnY q ˘ npθq ¯ , J6 “ p1 ´ pρnpθqqLApρnY nqpθq, where we denote A the constant matrix A “ ∇Xnp0q, Mp∇Xnq is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14), RpXnq in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18), LA in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='32), pxn “ x Xnp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The bound for J6 follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8 together with Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12, }J6}C1pR2q ď CpR, |X|˚, }∇S2X}C0pS2qq}TFpAq∇pρnY nq}C0pR2q, thus, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20) }J6}C1pR2q ď CpR, |X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2pρnY q}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10 with Z “ TSp∇S2Xq∇S2pρnY q provides the following bound for J4: }J4}CγpR2q ď Cp|X|˚, }∇S2X}C0pS2qq ` p1 ` }∇S2X}CγpS2qq}TSp∇S2Xq∇S2pρnY q}C0pS2q ` εpRq}TSp∇S2Xq∇S2pρnY q}CγpS2q ˘ , where εpRq Ñ 0 as R Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, }J4}CγpR2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C2q ´ εpRq}∇S2pρnY q}CγpS2q ` p1 ` }∇S2X}CγpS2qq}∇S2pρnY q}C0pS2q ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We proceed with J5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Recalling the expression for TS (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8), we have pTSp∇S2Xq∇S2pρnY qqn,lipηq “ pTSp∇S2Xq∇S2pρnY q ˝ x Xnql,ipηq “ pdetppgpηqqq´ 1 2 B Bηr pρnYn,qqpηqB p Xm Bηr pηq ´T pλnpηqq λnpηq δlqδim ` ` T 1pλnpηqq ´ T pλnpηq λnpηq ˘ BXn,l Bηj pηq Bx Xi Bηj pηq BXn,q Bηp pηq Bx Xm Bηp pηq pλnpηqq2detppgpηqq ¯ , WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 63 with λnpηq given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Substituting into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21) pMpAqpTSp∇S2Xq∇S2pρnY qqnqkpθq “ ´ ż R2 mm,k,lpθ, ηq B p Xi Bηm pηqpTSp∇S2Xpηqq∇S2pρnY nqpηqql,idη1dη2 “ ´ ż R2 mi,k,lpθ, ηqp ˜TSqipqlpηq B Bηp pρnYn,qqpηqdη1dη2 “ ˜ MpAqp ˜TS∇pρnY nqqpθq, where we denote (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22) p ˜TSp∇Xqqipqlpηq“ T pλnpηqq λnpηq δpiδql` ` T 1pλnpηqq´ T pλnpηq λnpηqq ˘ BXn,l Bηi pηq BXl,q Bηp pηq pλnpηqq2a detppgpηqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus we can write J5 “ pρnpθq ˜ MpAqppTF pAq ´ ˜TSp∇Xqq∇pρnY nqqpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3 with Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12 gives that }J5}CγpR2q ďCp|X˚, }∇S2X}C0pS2qq}pTF pAq ´ ˜TSp∇Xqq∇pρnY nq}CγpR2q, and since TF pAq “ ˜TSp∇Xp0qqp0q, we obtain }J5}CγpR2q ď Cp|X˚, }∇S2X}C0pS2q, }T }C2q ´ εpRq}∇S2pρnY q}CγpS2q ` }∇S2X}CγpS2q}∇S2pρnY q}C0pS2q ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we continue from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='19), }I1,1 n }CγpS2q ě }J3 n}CγpR2q ´ }J4 n}CγpR2q ´ }J5 n}CγpR2q ´ }J6 n}CγpR2q, so inserting back the bounds for J4 n, J5 n, and J6 n, we have that }I1,1 n }CγpS2q ě }J3 n}CγpR2q ´ Cp|X|˚, }∇S2X}C0pS2qqεpRq}∇S2pρnY q}CγpS2q ´ Cp|X|˚, }∇S2X}C0pS2qq}∇S2X}CγpS2q}∇S2pρnY q}C0pS2q ´ CpR, |X|˚, }∇S2X}C0pS2qq}∇S2pρnY q}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we use the frozen-coefficient estimate in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13 for J3 n (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We first interpolate the inequalities in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11 to control the lower-order terms, J3 n “ pz ´ LAqpρnY nqpθq, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='23) }J3 n}CγpR2q ě C|z|}ρnY n}CγpR2q ` C}ρnY n}C1,γpR2q ` C|z|1´σ}ρnY n}Cγ`σpR2q, where σ P r0, 1s is chosen so that 1 ă γ ` σ ă 1 ` γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, we have }I1,1 n }CγpR2q ě C|z|}ρnY }CγpS2q ` C}ρnY }C1,γpS2q ` C|z|1´σ}ρnY }Cγ`σpS2q ´ Cp|X|˚, }∇S2X}C0pS2qqεpRq}∇S2pρnY q}CγpS2q ´ Cp|X|˚, }∇S2X}C0pS2qq}∇S2X}CγpS2q}∇S2pρnY q}C0pS2q ´ CpR, |X|˚, }∇S2X}C0pS2qq}∇S2pρnY q}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 64 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN and taking R small enough, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24) }I1,1 n }CγpR2q ě C|z|}ρnY }CγpS2q ` C}ρnY }C1,γpS2q ` C|z|1´σ}ρnY }Cγ`σpS2q ´ CpR, |X|˚, }∇S2X}CγpS2qq}∇S2pρnY q}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, we go back to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16) and substitute the above bound together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='17) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24), }pz ´ S1pXqqY }CγpS2q ě ÿ n ´ C|z|}��nY }CγpS2q ` C}ρnY }C1,γpS2q ` C|z|1´σ}ρnY }Cγ`σpS2q ¯ ´ CpR, |X|˚, }∇S2X}CγpS2qq}∇S2Y }C0pS2q ´ Cp|X|˚, }∇S2X}CγpS2qq}∇S2ρn}CγpS2qp}Y }CγpS2q ` }∇S2Y }C0pS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Plugging this inequality in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13), and then using the triangle inequality and the fact that Cα ãÑ Cβ for α ě β, we obtain }pz ´ SpXqqY }CγpS2q ě C|z|}Y }CγpS2q ` C}Y }C1,γpS2q ` C|z|1´σ}Y }Cγ`σpS2q ´ CpR, |X|˚, }∇S2X}CγpS2qq}Y }Cγ`σpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, by moving the sector if necessary to make |z| big, we conclude the result }pz ´ SpXqqY }CγpS2q ě C|z|}Y }CγpS2q ` C}Y }C1,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The Gˆateaux derivative of F at any X P Om, SpXq (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2), generates an analytic semigroup on the space h0,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We need to prove that the operator SpXq is sectorial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', that there exists a sector such that for any z in the sector }pz ´ SpXqq´1Y }hγpS2q ď C |z|}Y }hγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since the norm on little H¨older spaces hγpS2q is the same as in the usual H¨older spaces CγpS2q, from the previous Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5 we are left to prove that the operator pz ´ SpXqq is invertible from hγpS2q to h1,γpS2q for any z in the sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Similarly as we did in Section 6, define the following family of operators SαpXq, α P r0, 1s, SαpXqY ppxq “ ´α ż S2∇S2 ´ G1pXppxq´Xppyqq`αG2pXppxq´Xppyqq ¯ ¨pTSp∇S2Xq∇S2Y ppyqqdpy ´p1 ´ αq ż S2∇S2 ´ G1pXppxq´Xppyqq`αG2pXppxq´Xppyqq ¯ ¨ ∇S2Y ppyqdpy ` αS2pXqY ppxq, with Gαpxq “ 1 8π pG1pxq ` αG2pxqq , x “ px1, x2, x3q, pG1qi,jpxq “ δij |x|, pG2qi,jpxq “ xixj |x|3 , WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 65 and TS given in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In particular, SpXq “ S1pXq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Propositions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5 hold analogously for SαpXq, as all the remainder estimates were always done inde- pendently for each part of the kernel G and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13 already included the parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In particular, for all α P r0, 1s, it holds that 1 C }Y }C1,γpS2q ě }pz ´ SαpXqqY }CγpS2q ě C}Y }C1,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, by the method of continuity, it suffices to show that the inverse of pz´S0pXqq exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Additionally, define a new family of operators S0,σpXq, σ P r0, 1s, as follows S0,σpXqY ppxq “ ´ ż S2∇S2 ´ p1´σqG1ppx´ pyq`σG1pXppxq´Xppyqq ¯ ¨ ∇S2Y ppyqdpy, so that S0,1pXq “ S0pXq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, taking into account (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='33), it is clear that the following bound holds for all σ P r0, 1s, 1 C }Y }C1,γpS2q ě }pz ´ S0,σpXqqY }CγpS2q ě C}Y }C1,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, by the method of continuity again we just need to show that pz ´ S0,0pXqq is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since the range is closed, it suffices to show that it is also dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The operator S0,0pXq is linear and explicit, so we can compute its eigenspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since S0,0pXqY “ 1 8π ż S2 1 |px ´ py|∆S2Y ppyqdpy, we only have to check a component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' From [21], a single layer potential u pxq of g ppyq with u pxq “ 1 4π ż S2 1 |x ´ py|g ppyq dpy can be transformed into a harmonic problem with ∆u “ 0 in R3zS2 �u� “ 0, �∇u ¨ n� “ g on S2 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='25) If we denote the standard spherical coordinate system pr, θ, ϕq, where r is the radial coordinate, θ is the polar angle, and ϕ is the azimuthal angle, then for the harmonic equation on R3zS2, by separation of variables [15], we obtain some solutions uℓm pr, θ, ϕq with l ě 0 and |m| ď ℓ : uℓm pr, θ, ϕq “ " ArℓYℓm, |r| ă 1 Br´pℓ`1qYℓm, |r| ą 1 where Yl,m pθ, ϕq is the usual spherical harmonic function of degree l and order m, which satisfies the following equation: (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='26) ∆S2Yℓm “ 1 sin θ B Bθ ˆ sin θBYℓm Bθ ˙ ` 1 sin2 θ B2Yℓm Bϕ2 “ ´ℓpℓ ` 1qYℓm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By plugging uℓm into (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='25), we obtain uℓm “ 1 2ℓ ` 1Yℓm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, combining (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='26), S0,0pXqYℓ,m “ ´ ℓpℓ ` 1q 2p2ℓ ` 1qYℓ,m, ℓ ě 0, |m| ď ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 66 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Finally, since finite linear combinations of Yl,m are dense in C8pS2q, we conclude the existence of the inverse pz ´ S0,0pXqq´1 : hγpS2q Ñ h1,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Higher Regularity Following the notation in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1, we recall that BX Bt ppxq “ NpXqpT p∇S2Xqqppxq, where we denote, with T given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4), T p∇S2Xq “ T p|∇S2X|q∇S2X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We localize using the partition tρnu (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4), and linearize T at Xppxnq, B BtpρnXqppxq “ ρnppxqNpXqpTSp∇S2Xppxnqq∇S2Xqqppxq ` ρnppxqNpXq ´ T p∇S2Xq´TSp∇S2Xppxnqq∇S2X ¯ ppxq, where we recall that TSp∇S2Xq∇S2Y “ d dsT p∇S2pX `sY qq|s“0 was given in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, we introduce the commutators, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) B BtpρnXqppxq “ NpXqpTSp∇S2Xppxnqq∇S2pρnXqqppxq ` NpXq ´ ρn ` T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X ˘¯ ppxq ` rρn, NpXqspTSp∇S2Xppxnqq∇S2Xqppxq ´ NpXqpTSp∇S2XppxnqqX∇S2ρnqppxq ` rρn, NpXqs ´ T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X ¯ ppxq, and we move to stereographic coordinates to introduce the frozen-coefficient (at t “ 0, px “ pxn) operator and the cutoff pρn (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) B BtpρnXnqpθq “ LA0pρnXnqpθq ` 7ÿ j“1 f jpXqpθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' with f 1pXqpθq “ NpXq ` ρn ` T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X ˘˘ npθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 2pXqpθq “ pρnpθq “ NpXq ´ MpAq ‰ pTSp∇S2Xppxnqq∇S2pρnXqqnpθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 3pXqpθq “ pρnpθq ` MpAqpTSp∇S2Xppxnqq∇S2pρnXqqnpθq´LApρnXnqpθq ˘ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 4pXqpθq “ p1 ´ pρnpθqqNpXqpTSp∇S2Xppxnqq∇S2pρnXqqnpθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 5pXqpθq “ ´p1 ´ pρnpθqqLApρnXnqpθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 6pXqpθq “ rLA ´ LA0spρnXnqpθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' and f 7pXqpθq “ ´NpXqpTSp∇S2XppxnqqX∇S2ρnqpx Xnpθqq ` rρn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' NpXqsT p∇S2Xqpx Xnpθqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' where A0 “ ∇X0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='np0q and A “ ∇Xnp0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 67 Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let X be the solution to the Peskin problem with initial data X0 P h1,γpS2q constructed in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, for any α P p0, 1q, it holds that X P C1pp0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C3,αpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, for any 3 ď n P N and α P p0, 1q, assuming that T P Cn,α, it holds that X P C1pp0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Cn`1,βpS2qq, for any β ă α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The main difficulty is to show the smoothing in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In fact, assume we have the higher regularity information X P L8p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Cn`1,αpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2 states that BtX P C0pr0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' CγpS2qq, and using the equation together with X P L8p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Cn`1,αpS2qq, it is straightforward to see that BtX P L8p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Cn,αpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, to get the continuity in time for the higher regularity, it suffices to inter- polate taking into account the higher regularity bounds and the continuity in the lower norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We proceed to show the smoothing in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will consider the following mollified version of the system (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) B BtpρnXδ nqpθq “ LA0pρnXδ nqpθq ` 7ÿ j“1 Jδf jpXδqpθq, with mollified initial data Xδ 0,npθq “ JδX0,npθq, where Jδ is the standard mollifier by convolution with a Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Our main goal is to obtain uniform in δ bounds for Xδ in L8p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Cn,αpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In fact, by construction, Xδ is smooth, and it is not hard to show that the limit of tXδu in L8p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C1,γpS2qq is given by the solution X in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, by interpolation and using the uniform bounds, we would conclude that X P L8p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Cn,βpS2qq for any β ă α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We thus proceed to obtain the uniform bounds first, and show the convergence Xδ Ñ X at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We use the semigroup etLA0 to write ρnXδ nptq in Duhamel form: (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) ρnXδ nptq “ ept´t0qLA0 pρnXδ npt0qq ` 7ÿ j“1 ż t t0 ept´τqLA0Jδf jpXδqpτqdτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In the following, we will repeatedly use the estimates in Propositions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For simplicity of notation, we will drop the index δ and the mollifier Jδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Improving regularity to C1,αpS2q: We proceed to obtain bounds in Cα, α P p0, 1q, for the terms f j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will be denoting C “ Cp|X|˚, }X}C1pS2q, }T }C2q, CpRq “ Cp|X|˚, }X}C1pS2q, }T }C2, Rq in the bounds that follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6 gives that }f1}CαpR2q ďC}ρnpT p∇S2Xq´TSp∇S2Xppxnqq∇S2Xq}CαpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We note that I :“ T p∇S2Xppx1qq´TSp∇S2Xppxnqq∇S2Xppx1q ´ T p∇S2Xppx2qq`TSp∇S2Xppxnqq∇S2Xppx2q “ T p∇S2Xppx1qq´Tp∇S2Xppx2qq´TSp∇S2Xppxnqqp∇S2Xppx1q´∇S2Xppx2qq, thus |I| “ | ż 1 0 ` TSps∇S2Xppx1q ` p1 ´ sq∇S2Xppx2qq ´ TSp∇S2Xppxnqq ˘ ds ˆ p∇S2Xppx1q´∇S2Xppx2qq| ď C maxt|∇S2Xppx1q´∇S2Xppxnq|, |∇S2Xppx2q´∇S2Xppxnqu ˆ |∇S2Xppx1q´∇S2Xppx2q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 68 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Hence, thanks to the presence of ρn, we obtain (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) }f 1}CαpR2q ďCεpRq}ρn}CαpS2q}∇S2X}CαpB pxn,2RXS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) }f 2}CαpS2q ď C ´ εpRq}∇S2pρnXq}CαpS2q ` }∇S2X}C α 2 pB5RppxnqXS2q}∇S2pρnXq}C α 2 pS2q ¯ , while f 3 is identically zero (see (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='32)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7 provides the estimate for f 4, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) }f 4}CαpS2q ď CpRq}∇S2pρnXq}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, by Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) }f 7}CαpS2q ď C}∇S2ρn}CαpS2q, and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) }f 5}CαpS2q ď C}∇S2pρnXq}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, by writing rLA ´ LA0spρnXnqpθq “ r ˜ MpAq ´ ˜ MpA0qspTF pAq∇pρnXnqqpθq ` ˜ MpA0qppTF pAq ´ TF pA0qq∇pρnXnqqpθq, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4 yields that (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) }f 6}CαpS2q ď C}A ´ A0}}∇S2pρnXq}CαpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We thus see from (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) and Propositions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15 that we can bootstrap to get that X P L8p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C1,αpS2qq for all α P p0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In fact, consider the case γ ă 1 2 and take α such that γ ă α ` 1 2 ď 2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, with 0 ă ǫ ď γ ´ α arbitrarily small, and substituting the bounds for f j, we obtain }ρnX}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpS2qq ď C}ρnp0qX0}C1`α`ǫpS2qq ` C 7ÿ j“1 }f j}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 1 2 `αpR2qq ď C}ρnp0qX0}C1`γpS2q ` CpR, T q ` }X}2 L4p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C1` 1 4 ` α 2 pS2qq ` CεpR, T q ` }X}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpB5RppxnqXS2qq ` }ρnX}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpS2q ˘ , and so (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) }ρnX}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpS2qq ď C}ρnp0qX0}C1`γpS2q ` CpR, T q ` CεpR, T q}X}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpB5RppxnqXS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Now, we can write }X}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpB5RppxnqXS2qq ď } ÿ mPMn ρmX}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpB5RppxnqXS2qq, where the cardinal number |Mn| can be picked independent of R and n, since the radius of the support of ρn and B5Rppxnq X S2 are comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, adding in n in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) we obtain ÿ n }ρnX}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpS2qq ď CpR, T q ` CεpR, T q|Mn| ÿ n }ρnX}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpS2qq, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 69 hence we conclude that }X}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpS2qq ď ÿ n }ρnX}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpS2qq ď CpR, T q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In particular, choosing α “ 2γ ´ 1 2, this uniform bound allows us to conclude that X P L2p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C1`2γpS2qq, and thus Xptq P C1`2γpS2q for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t P p0, T q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Now, pick t0 P p0, T q arbitrarily close to 0 and such that Xpt0q P C1`2γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It is clear that we can repeat the process to find t1 ą t0 such that Xpt1q P C1,αpS2q for any α P p0, 1q (the case γ ą 1 2 follows in one step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Starting at t1, we find that }ρnXptq}C1,αpS2q ď }ρnXpt1q}C1,αpS2q ` C sup t1ďτďt 7ÿ m“1 }fm}CαpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We can thus take the supremum in t P pt1, T q and use the previous estimates on f m to conclude that X P L8pt1, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C1,αpS2qq for any t1 ą 0 and any α P p0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Higher regularity: To study further smoothing, we first show that we can move derivatives in px to derivatives in py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In fact, denoting ∆X “ Xppxq ´ Xppyq, ∇S2NpXqY ppxq “ “ ´ ż S2 ∇S2,px∇S2, pyGp∆Xq ¨ ∆∇S2Y dpy “ ´ ż S2 ´ ´ ∇S2, py∇S2, pyGp∆Xq`∇S2, py ` ∇S2,pxGp∆Xq`∇S2, pyGp∆Xq ˘¯ ¨∆∇S2Y dpy, so further integration by parts gives that (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) ∇S2NpXqY ppxq “ NpXq∇S2Y ppxq ´ ż S2 ∇S2, py ` ∇S2,pxGp∆Xq`∇S2, pyGp∆Xq ˘ ¨ ∆∇S2Y ppyqdpy “ NpXq∇S2Y ppxq ´ ż S2 ∇S2, py ´ B Bxi Gp∆Xq ` ∇S2Xippxq ´ ∇S2Xippyq ˘¯ ¨ ∆∇S2Y ppyqdpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, we take a derivative in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) to get B Bt∇S2pρnXqppxq “ NpXq ` ∇S2 ` TSp∇S2Xppxnqq∇S2pρnXq ˘˘ ppxq ` NpXq ´ ∇S2` ρn ` T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X ˘˘¯ ppxq ´ ż S2 ∇S2, py ´ B Bxi Gp∆Xq ` ∇S2Xippxq ´ ∇S2Xippyq ˘¯ ¨ ˆ ∆ ` TSp∇S2Xppxnqq∇S2pρnXq ˘ ppyqdpy ´ ż S2 ∇S2, py ´ B Bxi Gp∆Xq ` ∇S2Xippxq ´ ∇S2Xippyq ˘¯ ¨ ˆ ∆ ` ρn ` T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X ˘ ppyqdpy ` ∇S2rρn, NpXqsT p∇S2Xqppxq ´ ∇S2NpXqpTSp∇S2XppxnqqX∇S2ρnqppxqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 70 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN We introduce the frozen-coefficient operator and the cutoff pρn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' B Bt∇S2pρnXnqpθq “ LA1p∇S2pρnXq ˘ npθq ` 8ÿ j“1 f jpθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 1pθq “ NpXq ´ ∇S2` ρn ` T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X ˘˘¯ npθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 2pθq “ pρnpθqrNpXq ´ MpAqs `` TSp∇S2Xppxnqq∇S2∇S2pρnXq ˘˘ npθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 3pθq“ pρnpθq ´ MpAqpTSp∇S2Xppxnqq∇S2∇S2pρnXqqnpθq´LAp∇S2pρnXqqnpθq ¯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 4pθq “ p1´pρnpθqqNpXq ` TSp∇S2Xppxnqq∇S2∇S2pρnXq ˘ npθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 5pθq “ p1´pρnpθqqLAp∇S2pρnXqqnpθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 6pθq “ rLA ´ LA1s ` ∇S2pρnXq ˘ npθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 7pθq “ ∇S2rρn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' NpXqsT p∇S2Xqpx Xnpθqq ´ ∇S2NpXqpTSp∇S2XppxnqqX∇S2ρnqpx Xnpθqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' and f 8pθq “ f 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1pθq ` f 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2pθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' with f 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1pθq “ ´ ż S2 ∇S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' py ´ B Bxi GpXpx Xpθq ´ Xppyqq ` ∇S2Xipx Xpθqq ´ ∇S2Xippyq ˘¯ ¨ ˆ ∆ ` TSp∇S2Xppxnqq∇S2pρnXq ˘ ppyqdpy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2pθq “ ´ ż S2 ∇S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' py ´ B Bxi GpXpx Xpθqq ´ Xppyqq ` ∇S2Xipx Xpθqq ´ ∇S2Xippyq ˘¯ ¨ ˆ ∆ ` ρn ` T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X ˘ ppyqdpy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' and A1 “ ∇Xnp0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A “ ∇Xnp0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, we proceed as we previously did in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) ∇S2pρnXnqptq “ ept´t0qLA1p∇S2pρnXnqpt0qq ` 8ÿ j“1 ż t t0 ept´τqLA1f jpτqdτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, to bootstrap and get C2,α regularity we need to use Propositions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14- 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15 and obtain Cα estimates for the forced terms above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18) in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10 gives that }f 2}CαpS2q ď C ´ εpRq}∇2 S2pρnXq}CαpS2q ` }∇S2X}CαpS2q}∇2 S2pρnXq}C0pS2q ` }∇2 S2pρnXq}C0pS2q ¯ , while f 3 ” 0, and Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8 provide that }f 4}CαpS2q ` }f 5}CαpS2q ď CpRq}∇2 S2X}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' As done before in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10), we have that }f 6}CαpS2q ď C}A ´ A1}}∇2 S2pρnXq}CαpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We interpolate the C2pS2q norm followed by Young’s inequality to get a small coefficient for the higher regularity part: }∇2 S2pρnXq}C0pS2q ď Cpεq}∇S2pρnXq}C1´αpS2q ` ε}∇2 S2pρnXq}CαpS2q, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 71 so that 6ÿ j“2 }f j}CαpS2q ď CεpR, ∆tq}∇2 S2pρnXq}CαpS2q`CpRq}∇S2pρnXq}C1´αpS2q, where from now on the constants C and CpR, ∆tq, ∆t “ t ´ t1, also depend on the controlled norm }X}L8p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C1,maxtα,1´αupS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, }f 1}CαpS2q ď Cε}∇S2X}CαpS2q ` C}ρn ` TSp∇S2Xq∇2 S2X ´ TSp∇S2ppxnqq∇2 S2X ˘ }CαpS2q ď C ` C}∇S2X}CαpS2q}∇2 S2X}C0pB pxn,2RXS2q ` CεpRq}∇2 S2X}CαpB pxn,2RXS2q, so by interpolation again }f 1}CαpS2q ď CpRq ` CεpRq}∇2 S2X}CαpB pxn,2RXS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The term f 8 is lower order, and thus we can control it using interpolation once more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In fact, taking the derivative in the kernel, we have for f 8,1 f 8,1ppxq “ ż S2 B Bxj B Bxi Gp∆Xq∇S2Xjppyq∆∇S2Xi ¨ ∆ ` TSp∇S2Xppxnqq∇S2pρnXq ˘ dpy ´ ż S2 B Bxi Gp∆Xq∇2 S2Xippyq ¨ ∆ ` TSp∇S2Xppxnqq∇S2pρnXq ˘ dpy, and therefore, proceeding as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9, we obtain }f 8,1}CαpS2q ď C ` C}∇2 S2X}C0pB pxn,2RXS2q ď CpRq ` CεpRq}∇2 S2X}CαpB2RppxqXS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The estimate for f 8,2 follows in the same manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, we estimate the commu- tator terms, f 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Using (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' we write f 7ppxq “ f 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1ppxq ` f 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2ppxq ` f 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3ppxq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' with f 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1ppxq “ ´rNpXq∇S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ρnsT p∇S2Xqppxq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2ppxq “ ż S2 ∇S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' py ´ B Bxi Gp∆Xq ` ∇S2Xippxq ´ ∇S2Xippyq ˘¯ ¨ ∆ ` ρnT p∇S2Xqppyq ˘ dpy ´ ρnppxq ż S2 ∇S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' py ´ B Bxi Gp∆Xq ` ∇S2Xippxq ´ ∇S2Xippyq ˘¯ ¨ ∆ ` T p∇S2Xqppyq ˘ dpy ` ż S2∇S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' py ´ B Bxi Gp∆Xq ` ∇S2Xippxq´∇S2Xippyq ˘¯ ¨∆ ` TSp∇S2XppxnqqX∇S2ρnppyq ˘ dpy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' and f 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3ppxq “ ´NpXq ` TSp∇S2Xppxnqq∇S2pX∇S2ρnq ˘ ppxq ` ∇S2ρnNpXqpT p∇S2Xqqppxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The term f 7,3 is lower order and it only requires C1,αpS2q regularity for X, while the estimate for f 7,2 follows taking the derivative of the kernel, as done for f 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We get that }f 7,2}CαpS2q ď C ` C}∇2 S2X}C0pB pxn,2RXS2q ` C}∇2 S2X}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By interpolation, }f 7,2}CαpS2q ` }f 7,3}CαpS2q ď Cp˜εq ` C˜ε}∇2 S2X}CαpS2q, 72 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN with ˜ε ą 0 to be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The term f 7,1 is written as follows: f 7,1ppxq “ ´ ż S2 ∇S2, pyGpXppxq´Xppyqq ¨ ` ρnppxq∇S2T p∇S2Xppyqq ´ ∇S2 ` ρnppyqT p∇S2Xppyqq ˘˘ dpy “ rρn, NpXqs∇S2T p∇S2Xqppxq ` ż S2 ∇S2, pyGpXppxq´Xppyqq ¨ ∇S2ρnppyqT p∇S2Xppyqqdpy, hence, by Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6, we have that }f 7,1}CαpS2q ď C}∇S2ρn}C1,αpS2q ` C}∇S2ρn}C0pS2q}∇2 S2X}C0pS2q, and, by interpolation, }f 7,1}CαpS2q ď Cp˜εq ` C˜ε}∇2 S2X}CαpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we have that for any t1 ą 0 and α P p0, 1 2q, }∇S2pρnXqptq}L2pt1,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C1,αpS2qq ď C}ρnpt1qXpt1q}C 3 2 `α´ǫ ` 7ÿ m“1 }f mpτq}L2pt1,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='CαpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, introducing the estimates above for f j and summing in n, we take the partition so that εpRq is small enough and then choose ˜ε small enough (depending on the partition ρn), to obtain that X P L2pt1, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C2,αpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, we can take t2 ą t1 so that Xpt2q P C2,αpS2q and use (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) to conclude that X P L8pt2, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C2,αpS2qq for any t2 ą 0, α P p0, 1 2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Now, starting with the upgraded regularity and repeating the same steps with no changes, we conclude that X P L8pt2, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C2,αpS2qq for any t2 ą 0, α P p0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It is not difficult to show by induction that an analogous formula to (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) holds for higher derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, by repeating the steps above one can continue the bootstrapping argument, concluding that for any n P N, X P L8p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Cn,αpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Xδ Ñ X in L8p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C1,γpS2qq: We write the difference ∆δX :“ Xδ´X as follows: ρn∆δXnptq “ etLA0 ` ρn∆δX0,n ˘ ` 7ÿ j“1 ż t 0 ept´τqLA0 ´ pJδ ´ 1qf jpXδqpτq ` f jpXδqpτq ´ f jpXqpτq ¯ dτ, with f jpXq given in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) }ρn∆δXn}L8p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C1,γpR2qq ď C}ρn∆δX0,n}C1,γpR2q ` C 7ÿ j“1 }pJδ ´ 1qf jpXδq}L8p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='CγpR2qq ` C sup tPr0,T s 7ÿ j“1 } ż t 0 ept´τqLA0pf jpXδq ´ f jpXqqpτq}CγpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since X0 P h1,γpS2q and f jpXδq P hγpS2q, the first two terms converge to zero as δ Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the third term, we need to show that it can be absorbed by the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' As in the previous arguments, we will show that for the quasilinear terms we WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 73 can find a small coefficient, while for the lower order ones we will take advantage of the extra regularity via (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) to get a small coefficient for T small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' From previous estimates we immediately get that for j “ 4, 5, 7 and 0 ă ǫ ă 1´γ, }f jpXδq ´ f jpXq}Cγ`ǫpS2q ď C}∆δX}C1pS2q, hence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) gives that sup tPr0,T s ÿ j“4,5,7 } ż t 0 ept´τqLA0pf jpXδq ´ f jpXqqpτq}CγpR2q ď C T ǫ}∆δX}C1pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Also, for f 6, }f 6pXδq ´ f 6pXq}CγpS2q ď C}A ´ A0}}∇S2pρn∆δXq}CγpS2q ` C}∆δX}C1pS2q}ρnXδ}C1,γpS2q, so that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15) give } ż t 0 ept´τqLA0pf 6pXδq ´ f 6pXqqpτq}L8p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='CγpR2qq ď CεpT q}ρn∆δX}C1,γpS2q ` CT ε}∆δX}C1pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, we proceed with the term f 1: pf 1pXδq ´ f 1pXqqpθq “ I1 ` I2, with I1pθq “ pNpXδq ´ NpXqq ` ρn ` T p∇S2Xδq ´ TSp∇S2Xδppxnqq∇S2Xδ˘˘ npθq I2pθq “ NpXq ´ ρn ´ T p∇S2Xδq ´ TSp∇S2Xδppxnqq∇S2Xδ ´ T p∇S2Xq ` TSp∇S2Xppxnqq∇S2X ¯¯ npθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The first term is estimated easily from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6 by noticing that one can always extract ∆δX “ Xδ ´X from the difference of the kernels (similarly as done in the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15) }I1}CγpR2q ďC}∇S2∆δX}C0pS2q}ρnpT p∇S2Xδq´TSp∇S2Xδppxnqq∇S2Xδq}CγpS2q ď CεpRq}∇S2∆δX}C0pS2q}∇S2X}CγpB pxn,2RXS2q, where we have used (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) in the second step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6 gives that (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16) }I2}CγpR2q “ C}ρn ´ T p∇S2Xδq ´ TSp∇S2Xδppxnqq∇S2Xδ ´ T p∇S2Xq ` TSp∇S2Xppxnqq∇S2X ¯ }CγpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Denote JpY qppxq “ T p∇S2Y ppxqq ´ TSp∇S2Y ppxnqq∇S2Y ppxq, so that we can write JpXδppx1qq ´ JpXδppx2qq “ p∇S2Xδppx1q ´ ∇S2Xδppx2qq ˆ ż 1 0 ` TSps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ´ TSp∇S2Xδppxnqq ˘ ds1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 74 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Then, JpXδppx1qq ´ JpXδppx2qq ´ JpXppx1qq ` JpXppx2qq “ J1 ` J2, with J1 “ p∇S2∆δXppx1q ´ ∇S2∆δXppx2qq ˆ ż 1 0 ` TSps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ´ TSp∇S2Xδppxnqq ˘ ds1, and J2 “ p∇S2Xppx1q ´ ∇S2Xppx2qq ˆ ´ ż 1 0 ` TSps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ´ TSp∇S2Xδppxnqq ˘ ds1 ´ ż 1 0 ` TSps1∇S2Xppx1q ` p1 ´ s1q∇S2Xppx2qq ´ TSp∇S2Xppxnqq ˘ ds1 ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It follows that |J1| ď C maxt|∇S2Xδppx1q´∇S2Xδppxnq|, |∇S2Xδppx2q´∇S2Xδppxnqu ˆ |∇S2∆δXppx1q´∇S2∆δXppx2q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We apply the mean-value theorem again in J2: ż 1 0 ` TSps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ´ TSp∇S2Xδppxnqq ˘ ds1 “ ż 1 0 ż 1 0 DTS ` s2ps1∇S2Xδppx1q`p1´s1q∇S2Xδppx2qq`p1´s2q∇S2Xδppxnq ˘ ds1ds2 ˆ ` s1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2q ´ ∇S2Xδppxnq ˘ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' hence adding and subtracting we obtain |J2| ď |J2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1| ` |J2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' with |J2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1| ď |∇S2Xppx1q ´ ∇S2Xppx2q| ˆ maxt|∇S2Xδppx1q´∇S2Xδppxnq|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' |∇S2Xδppx2q´∇S2Xδppxnq|u ˆ ˇˇˇDTSps2ps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ` p1 ´ s2q∇S2Xδppxnqq ´ DTSps2ps1∇S2Xppx1q ` p1 ´ s1q∇S2Xppx2qq ` p1 ´ s2q∇S2Xppxnqq ˇˇˇ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' |J2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2| ď |∇S2Xppx1q ´ ∇S2Xppx2q| ˆ |DTSps2ps1∇S2Xppx1q ` p1 ´ s1q∇S2Xppx2qq ` p1 ´ s2q∇S2Xppxnqq| ˆ maxt|∇S2∆δXppx1q´∇S2∆δXppxnq|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' |∇S2∆δXppx2q´∇S2∆δXppxnqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Going back to (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16), we thus conclude that }I2}CγpR2q ď CεpRq}∇S2∆δX}CγpB pxn,2RXS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Together with the bound for I1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15), we obtain the following estimate for f 1: } ż t 0 ept´τqLA0pf 1pXδq ´ f 1pXqqpτq}L8p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='CγpR2qq ď C T ǫ}∆δX}C1pS2q ` CεpRq}∆δX}C1pS2q}X}C1,γpB pxn,2RXS2q ` CεpRq}∆δX}C1,γpB pxn,2RXS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 75 The estimate for f 2 follows in the same way than those for f 1 and f 6, from which we conclude that }ρn∆δXn}L8p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C1,γpR2qq ď C}ρn∆δX0,n}C1,γpR2q ` C 7ÿ j“1 }pJδ ´ 1qf jpXδq}L8p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='CγpR2qq ` CT ε}∆δX}C1pS2q ` CεpRq}∆δX}C1pS2q}X}C1,γpB pxn,2RXS2q ` CpεpT q ` εpRqq}∆δX}C1,γpB px,2RXS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Taking R and T small enough, the last term is absorbed by the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, adding in n, we conclude that, for T and R small enough, the desired estimate holds }∆δX}L8p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C1,γpS2qq ďC}∆δX0}C1,γpS2q`C 7ÿ j“1 }pJδ´1qfjpXδq}L8p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='CγpR2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Besov Spaces and Fourier Multiplier Theorems In this section, we will proof Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' First, define Besov Spaces Bγ p,q by a dyadic decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Set a function ψ pξq P C8 pRnq s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ψ pξq “ " 1 |ξ| ď 1 0 |ξ| ě 2 and define φ pξq :“ ψ pξq ´ ψ p2ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, φ pξq P C8 pRnq and φ pξq “ 0, |ξ| ď 1 2, |ξ| ě 2, 8 ÿ j“´8 φ ` 2´jξ ˘ “ 1, |ξ| ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, the homogeneous dyadic blocks 9∆j are defined by 9∆jf pθq :“ F´1 ` φ ` 2´jξ ˘ Ff pξq ˘ pθq “ Kj ˚ f (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) where K pθq :“ F´1 pφ pξqq pθq and Kj pθq :“ F´1 ` φ ` 2´jξ ˘˘ pθq “ 2jnK ` 2jθ ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Now, given γ a real number and p, q ě 1, we may define homogeneous Besov spaces 9Bγ p,q pRnq with its seminorm ∥¨∥ 9Bγ p,qpRnq by ∥f∥ 9Bγ p,qpRnq :“ ˜ 8 ÿ j“´8 ˆ 2jγ ��� 9∆jf ��� LppRnq ˙q¸ 1 q , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) ∥f∥ 9Bγ p,8pRnq :“ sup jPZ ˆ 2jγ ��� 9∆jf ��� LppRnq ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) According to [27, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2] and [48, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2], we know for all 0 ă γ ă 1, ∥¨∥ 9Bγ p,qpRnq and �¨�CγpRnq are equivalent, so we only need to prove the Fourier multiplier theorem on 9Bγ p,q pRnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The proof is from [48, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given T a Fourier multiplier operator with multiplier m pξq P Cs pRnzt0uq X L8 pRnq, for s ą n 2 and for all |α| ď s, such that ��Bα ξ m pξq �� ď Cα |ξ|´|α| , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) 76 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN we first define a related kernel with λ ą 0 by Kλ pθq :“ F´1 pφ pξq m pλξqq pθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given m pξq satisfying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4), then Kλ pθq is bounded by ż Rn |Kλ pθq| dθ ď Cs,n,φDm, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) where Dm “ max|α|ďs Cα Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since there exist Cs s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' for all θ P Rn ´ 1 ` |θ|2¯s ď Cs ÿ |α|ďs |θα|2 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) we obtain ż Rn |Kλ pθq|2 ´ 1 ` |θ|2¯s dθ ďCs ÿ |α|ďs ż Rn |θαKλ pθq|2 dθ “Cn,s ÿ |α|ďs ż Rn ��Bα ξ pφ pξq m pλξqq ��2 dξ “Cn,s ÿ |α|ďs ż Rn ����� ÿ βďα ˆ α β ˙ ´ Bβ ξ φ ¯ pξq λ|α´β| ´ Bα´β ξ m ¯ pλξq ����� 2 dξ ďCn,sD2 m ÿ |α|ďs ż Rn ����� ÿ βďα ˆ α β ˙ ´ Bβ ξ φ ¯ pξq λ|α´β| |λξ|´|α´β| ����� 2 dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) supp pφq Ă ␣ ξ| 1 2 ď |ξ| ď 2 ( , so ż Rn ����� ÿ βďα ˆ α β ˙ ´ Bβ ξ φ ¯ pξq λ|α´β| |λξ|´|α´β| ����� 2 dξ “ ż 1 2 ď|ξ|ď2 ����� ÿ βďα ˆ α β ˙ ´ Bβ ξ φ ¯ pξq |ξ|´|α´β| ����� 2 dξ ď Cφ,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) Thus, ż Rn |Kλ pθq|2 ´ 1 ` |θ|2¯s dθ ď Cn,sD2 m ÿ |α|ďs Cφ,α ď Cs,n,φD2 m, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) and by Holder inequality, ż Rn |Kλ pθq| dθ ď ˆż Rn |Kλ pθq|2 ´ 1 ` |θ|2¯s dθ ˙ 1 2 ˆż Rn ´ 1 ` |θ|2¯´s dθ ˙ 1 2 ď a Cs,n,φDm ˆż Rn ´ 1 ` |θ|2¯´s dθ ˙ 1 2 ď Cs,n,φDm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) □ Next, we may use Kλ pθq and homogeneous Besov semi norm ∥¨∥ 9Bγ p,qpRnq to prove the Fourier multiplier theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 77 Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' First, set T j mf pθq :“ 9∆jTmf pθq “ F´1 ` φ ` 2´jξ ˘ m pξq Ff pξq ˘ pθq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) Since F´1 ` φ ` 2´jξ ˘ m pξq ˘ pθq “ 2njK2j ` 2jθ ˘ , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) ��T j mf pθq �� “ ��F´1 ` φ ` 2´jξ ˘ m pξq ˘ ˚ f pθq �� ď ��F´1 ` φ ` 2´jξ ˘ m pξq ˘ pθq �� L1pRnq ∥f∥L8pRnq ď ��2njK2j ` 2jθ ˘�� L1pRnq ∥f∥L8pRnq ď ∥K2j pθq∥L1pRnq ∥f∥L8pRnq ďCs,n,φDm ∥f∥L8pRnq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) Next, supp ` φ ` 2´jξ ˘˘ Ă ␣ ξ ˇˇ2j´1 ď |ξ| ď 2j`1 ( , so for all j ` 1 ď k ´ 1 or j ´ 1 ě k ` 1 φ ` 2´jξ ˘ φ ` 2´kξ ˘ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) Therefore, φ ` 2´jξ ˘ Ff pξq “ φ ` 2´jξ ˘ ´ F ´ 9∆j´1f ¯ pξq ` F ´ 9∆jf ¯ pξq ` F ´ 9∆j`1f ¯ pξq ¯ , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15) so we obtain T j mf pθq “ T j m ´ 9∆j´1f ¯ pθq ` T j m ´ 9∆jf ¯ pθq ` T j m ´ 9∆j`1f ¯ pθq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16) Finally, ∥Tmf∥ 9Bγ 8,8pRnq “ sup jPZ 2jγ ��T j mf �� L8pRnq ď sup jPZ 2jγ ���T j m ´ 9∆j´1f ¯��� L8pRnq ` sup jPZ 2jγ ���T j m ´ 9∆jf ¯��� L8pRnq ` sup jPZ 2jγ ���T j m ´ 9∆j`1f ¯��� L8pRnq ďCs,nDm ˆ sup jPZ 2jγ ��� 9∆j´1f ��� L8pRnq ` sup jPZ 2jγ ��� 9∆jf ��� L8pRnq ` sup jPZ 2jγ ��� 9∆j`1f ��� L8pRnq ˙ “Cs,nDm ` 2γ ` 1 ` 2´γ˘ ∥f∥ 9Bγ 8,8pRnq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='17) Since ∥¨∥ 9Bγ p,qpRnq and �¨�CγpRnq are equivalent, �Tmu�CγpRnq ď Cγ,s,nDm�u�CγpRnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18) □ Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Estimates for the semigroup e´tLApξq Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For all β “ β1β2 ¨ ¨ ¨ βk, there exists a matrix Pβ ´ ˆξ1, ˆξ2 ¯ of polyno- mials with degree deg pPβq ď 3 |β| ` 4 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' BβLA pξq “ 1 |ξ||β|´1 Pβ ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 2|β|`3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) 78 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN More specifically, Pβ ´ ˆξ1, ˆξ2 ¯ can be written as pPβqi1i2 ´ ˆξ1, ˆξ2 ¯ “ ÿ j1,j2ě0,j1`j2ď3|β|`4 cpβ,i1,i2q j1,j2 ˆ A, U, P, 1 detpBq, T , dT dλ , 1 λ2 ˙ ˆξj1 1 ˆξj2 2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) where cpβ,i1,i2q j1,j2 ˆ A, U, P, 1 detpBq, T , dT dλ , 1 λ2 ˙ “cpβ,i1,i2q j1,j2 ˆ A11, ¨ ¨ ¨ , A32, U11, ¨ ¨ ¨ , U32, P11, ¨ ¨ ¨ , P33, 1 detpBq, T λ , dT dλ , 1 λ2 ˙ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) is a polynomial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, ���Pβ ´ ˆξ1, ˆξ2 ¯��� C1pDAσ1,σ2q and ���Uˆξ ��� C1pDAσ1,σ2q are uniformly bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' there exists Cpβq σ1,σ2,T and Cpβq σ1,σ2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' for all ˆξ P S1, ���Pβ ´ ˆξ1, ˆξ2 ¯��� C1pDAσ1,σ2q ď Cpβq σ1,σ2,T , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) ���Uˆξ ��� C1pDAσ1,σ2q ď Cpβq σ1,σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since pFθGα,Aq pξq “ 1 |ξ| pFθGα,Aq pˆξq (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) “ 1 |ξ| pI ` αPq ���Uˆξ ��� 2 ´ αUˆξ b Uˆξ 4 detpBq ���Uˆξ ��� 3 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) and Zpξq “ |ξ|2 Zpˆξq where Zpˆξq is a matrix of polynomials with degree 2, LA pξq “ |ξ| P0 ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 3 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) where P0 “ pI ` αPq ���Uˆξ ��� 2 ´ αUˆξ b Uˆξ 4 detpBq ˜ T λ ˜ I ´ Aˆξ b Aˆξ λ2 ¸ ` dT dλ Aˆξ b Aˆξ λ2 ¸ , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) where the degree of P0 is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Obviously, P0 can be written as (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 79 When |β| “ 1, BLA Bξi pξq “ ξi |ξ| P0 ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 3 ` |ξ| ÿ j“1,2 B Bˆξj P0 ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 3 B Bξi ξj |ξ| “ ξi |ξ| P0 ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 3 ` |ξ| ÿ j“1,2 BP0 Bˆξj ���Uˆξ ��� 2 ´ 3P0 ´ U T Uˆξ ¯ j ���Uˆξ ��� 5 δij ´ ˆξi ˆξj |ξ| “ Pi ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 5 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) where Pi ´ ˆξ1, ˆξ2 ¯ “ ˆξiP0 ���Uˆξ ��� 2 ` ÿ j“1,2 ˜ BP0 Bˆξj ���Uˆξ ��� 2 ´ 3P0 ´ U T Uˆξ ¯ j ¸ ´ δij ´ ˆξi ˆξj ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) The degrees of all terms are at most 1 ` 4 ` 2 “ 3 ` 2 ` 2 “ 4 ` 1 ` 2 “ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since ���Uˆξ ��� 2 “ 2ÿ j1,j2“1 3ÿ k“1 Ukj1Ukj2 ˆξj1 ˆξj2, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) ´ U T Uˆξ ¯ j “ „ř3 k“1 UkjUk1 ˆξj ř3 k“1 UkjUk2 ˆξj \uf6be , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) and BP0 Bˆξj can be written as the form of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2), Pi ´ ˆξ1, ˆξ2 ¯ can be written as (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, the case |β| “ 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Suppose |β| ď k´1 holds, for ��¯β �� “ k, we may rewrite ¯β as ββk where |β| “ k´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, B ¯βLA pξq “ BβkBβLA pξq “ B Bξβk ¨ ˚ ˝ 1 |ξ||β|´1 Pβ ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 2|β|`3 ˛ ‹‚ “ ´ p|β| ´ 1q 1 |ξ||β| ˆξβk Pβ ���Uˆξ ��� 2|β|`3 ` 1 |ξ||β|´1 ÿ j“1,2 BPβ Bˆξj ���Uˆξ ��� 2 ´ p2 |β| ` 3q Pβ ´ U T Uˆξ ¯ j ���Uˆξ ��� 2|β|`5 δβkj ´ ˆξβk ˆξj |ξ| “ 1 |ξ|| ¯β| P¯β ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� ¯β`3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) 80 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN where Pˆβ “ ´ p|β| ´ 1q ˆξiPβ ���Uˆξ ��� 2 ` ÿ j“1,2 ˜ BPβ Bˆξj ���Uˆξ ��� 2 ´ ` 2 ��¯β �� ` 1 ˘ Pβ ´ U T Uˆξ ¯ j ¸ ´ δβkj ´ ˆξβk ˆξj ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15) The degrees of all terms are at most 1 ` p3 |β| ` 4q ` 2 “ p3 |β| ` 3q ` 2 ` 2 “ p3 |β| ` 4q ` 1 ` 2 “ 3 ��¯β �� ` 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Again, BPβ Bˆξj is still able to written as the form of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2), so the case ��¯β �� “ k holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By Induction, for all β, the formulas (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, in Pβ, since for all square matrix M, ∥M∥ ď ř |Mij|, we just need to estimate each element pPβqi1i2, ���pPβqi1i2 ´ ˆξ1, ˆξ2 ¯��� ď ÿ ����cpβ,i1,i2q j1,j2 ˆ A, U, P, 1 detpBq, T λ , dT dλ , 1 λ2 ˙���� (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16) and cpβ,i1,i2q j1,j2 ´ A, U, P, 1 detpBq, T λ , dT dλ , 1 λ2 ¯ is a form of a polynomial, so we may only check each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given A P DA, ��� ` AT A ˘´1��� , 1 detpBq ď 1 σ2 2 and σ1 ď λ ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2σ1, so all variables are bounded by σ1 and σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ���� BAT A BAij ���� ď 2λ ď 2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2σ1, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='17) ����� B ` AT A ˘´1 BAij ����� “ ���� ` AT A ˘´1 BAT A BAij ` AT A ˘´1 ���� ď 2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2σ1 σ4 2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18) so on DA, U, P are C1 functions and their derivatives are bounded by σ1 and σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Absolutely, ���Uˆξ ��� C1pDAσ1,σ2q ď Cpβq σ1,σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='19) Since ����� B det ` AT A ˘ BAij ����� ď λ2 ď 8σ2 1, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20) ���� B BAij 1 detpBq ���� “ ����� 1 2 detpBq3 B det ` AT A ˘ BAij ����� ď 8σ2 1 σ8 2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21) and ���� Bλ BAij ���� “ ���� Aij λ ���� ď 1 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22) on DA, 1 detpBq, T λ , dT dλ , 1 λ2 are also C1 functions and their derivatives are bounded by σ1 and σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, we obtain ���Pβ ´ ˆξ1, ˆξ2 ¯��� C1pDAσ1,σ2q ď Cpβq σ1,σ2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='23) □ WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 81 Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given A P DAσ1,σ2 and ϕ pξq “ ϕ p|ξ|q, a cutting and decreasing respect |ξ| and supported in B p1q, and set K0 pθq :“ F´1 ” pz ` LAq´1 pξqϕpξq ı , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24) K1,j pθq :“ F´1 ” ξj pz ` LAq´1 pξqϕpξq ı .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='25) Then, for all z P Sω,δ, we have the following estimates ∥K0 pθq∥ ďCω,δ,σ1,σ2,T |z| 1 1 ` |θ|3 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='26) ∥K1,j pθq∥ ďCω,δ,σ1,σ2,T 1 1 ` |θ|4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='27) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For convenience, we define Hpξq :“ pz ` LAq´1 pξq First, since p1 ` |θ|3q ż R2 eiθ¨ξ pz ` LAq´1 pξqϕpξqdξ “ ż R2p1 ´ i θ |θ| ¨ iθ|θ|2qeiθ¨ξ pz ` LAq´1 pξqϕpξqdξ “ ż R2 „ p1 ´ i θ |θ| ¨ ∇ξ|∇ξ|2qeiθ¨ξ \uf6be pz ` LAq´1 pξqϕpξqdξ “ ż R2 eiθ¨ξ pz ` LAq´1 pξqϕpξq ` ieiθ¨ξ θ |θ| ¨ ∇ξ∆ξ ” pz ` LAq´1 pξqϕpξq ı dξ “ ż Bp1q eiθ¨ξ pz ` LAq´1 pξqϕpξq ` ieiθ¨ξ θ |θ| ¨ ∇ξ∆ξ ” pz ` LAq´1 pξqϕpξq ı dξ, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='28) By (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20), ����� ż Bp1q eiθ¨ξ pz ` LAq´1 pξqϕpξqdξ ����� ď Cδ,σ1,σ2,T ∥ϕ∥C0 |z| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='29) Next, we compute B Bξj ∆ξ ” pz ` LAq´1 ϕ ı “ B Bξj ∆ξ pz ` LAq´1 ϕ ` ∆ξ pz ` LAq´1 B Bξj ϕ ` 2 B Bξj ∇ξ pz ` LAq´1 ¨ ∇ξϕ `2∇ξ pz ` LAq´1 ¨ B Bξj ∇ξϕ ` B Bξj pz ` LAq´1 ∆ξϕ ` pz ` LAq´1 B Bξj ∆ξϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='30) Since ϕ is smooth, we may estimate all of the terms except the first by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Obviously, for the last term, �����i θj |θ| ż Bp1q eiθ¨ξ pz ` LAq´1 ˆ B Bξj ∆ξϕ ˙ dξ ����� ď Cδ,σ1,σ2,T ∥ϕ∥C3 |z| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='31) 82 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Then, for all |α| “ 1, 2, |α| ` |β| “ 3 �����i θj |θ| ż Bp1q eiθ¨ξBα ξ pz ` LAq´1 pξqBβ ξ ϕpξqdξ ����� ď ∥ϕ∥C2 ż Bp1q ���Bα ξ pz ` LAq´1 pξq ��� dξ ďCδ,σ1,σ2,T |z|2 ∥ϕ∥C2 ż Bp1q |ξ|1´|α| dξ ď Cδ,σ1,σ2,T |z|2 ∥ϕ∥C2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='32) Now, for the first term, B Bξj ∆ξH “2 ÿ k“1,2 “ ´ pEjkk ` Ekjk ` Ekkjq ` ` Ekpjkq ` Epjkqk ˘‰ ` ` Ejpkkq ` Epkkqj ˘ ´ H B Bξj ∆ξLAH, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='33) where Eijk “H BLA Bξi H BLA Bξj H BLA Bξk H, Ekpjkq “H BLA Bξk H B2LA BξjBξk H, Ejpkkq “H BLA Bξj H∆ξLAH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since ��� BLA Bξj ��� ≲ 1 and ��� B2LA BξjBξk ��� ≲ |ξ|´1, ∥Eijk∥ ≲ 1 and ��Ekpjkq �� , ��Ejpkkq �� ≲ |ξ|´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, we only have to check pz ` LAq´1 B Bξj ∆ξLA pz ` LAq´1 ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' LA is an even function, so the term is an odd function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By LA pξq “ |ξ| LA ´ ˆξ ¯ and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10), �����i θj |θ| ż Bp1q eiθ¨ξ pz ` LA pξqq´1 B Bξj ∆ξLA pξq pz ` LA pξqq´1 ϕ pξq dξ ����� “ �����´ θj |θ| ż Bp1q sin pθ ¨ ξq pz ` LA pξqq´1 B Bξj ∆ξLA pξq pz ` LA pξqq´1 ϕ pξq dξ ����� ď ������ ż Bp1q sin ´ θ ¨ ˆξ |ξ| ¯ ´ z ` |ξ| LA ´ ˆξ ¯¯´1 Φp3q A,j ´ ˆξ ¯ |ξ|2 ´ z ` |ξ| LA ´ ˆξ ¯¯´1 ϕ p|ξ|q dξ ������ “ ������ ż S1 ż 1 0 ´ z ` rLA ´ ˆξ ¯¯´1 Φp3q A,j ´ ˆξ ¯ ´ z ` rLA ´ ˆξ ¯¯´1 ϕ prq sin ´ θ ¨ ˆξr ¯ r drdˆξ ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='34) WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 83 By Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4, we obtain for all ˆξ P S1 ������ ż 1 0 ´ z ` rLA ´ ˆξ ¯¯´1 Φp3q A,j ´ ˆξ ¯ ´ z ` rLA ´ ˆξ ¯¯´1 ϕ prq sin ´ θ ¨ ˆξr ¯ r dr ������ ď2 ���� ´ z ` rLA ´ ˆξ ¯¯´1 Φp3q A,j ´ ˆξ ¯ ´ z ` rLA ´ ˆξ ¯¯´1 ϕ prq ���� C1pr0,1sq ď4 ���Φp3q A,j ´ ˆξ ¯��� ∥ϕ prq∥C1pr0,1sq ���� ´ z ` rLA ´ ˆξ ¯¯´1���� 2 C1pr0,1sq ďCδ,σ1,σ2,T ∥ϕ prq∥C1pr0,1sq ˜ 1 |z| ` 1 |z|2 ¸2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='35) Therefore, ������ ż S1 ż 1 0 ´ z ` rLA ´ ˆξ ¯¯´1 Φp3q A,j ´ ˆξ ¯ ´ z ` rLA ´ ˆξ ¯¯´1 ϕ prq sin ´ θ ¨ ˆξr ¯ r drdˆξ ������ ďCδ,σ1,σ2,T ∥ϕ prq∥C1 ˜ 1 |z| ` 1 |z|2 ¸2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='36) Since 1 |z| ď Cω,δ if z P Sω,δ, ����� ż Bp1q eiθ¨ξ pz ` LAq´1 pξqϕpξq ` ieiθ¨ξ θ |θ| ¨ ∇ξ∆ξ ” pz ` LAq´1 pξqϕpξq ı dξ ����� ď ∥ϕ prq∥C3 4ÿ k“1 Cpkq δ,σ1,σ2,T 1 |z|k ďCω,δ,σ1,σ2,T ∥ϕ∥C3 |z| , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='37) and ���� ż R2 eiθ¨ξ pz ` LAq´1 pξqϕpξqdξ ���� ďCω,δ,σ1,σ2,T ∥ϕ∥C3 |z| 1 1 ` |θ|3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='38) Next, for K1,j, we use the same technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ` |θ|4qK1,j pθq becomes p1 ` |θ|4q ż R2 eiθ¨ξξj pz ` LAq´1 pξqϕpξqdξ “ ż R2 “ p1 ` |∇ξ|4qeiθ¨ξ‰ ξj pz ` LAq´1 pξqϕpξqdξ “ ż Bp1q eiθ¨ξξj pz ` LAq´1 pξqϕpξq ` eiθ¨ξ∆2 ξ ” ξj pz ` LAq´1 pξqϕpξq ı dξ, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='39) By (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20), ����� ż Bp1q eiθ¨ξξj pz ` LAq´1 pξqϕpξqdξ ����� ď Cδ,σ1,σ2,T ∥ϕ∥C0 |z| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='40) 84 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Next, ∆2 ξ ” ξj pz ` LAq´1 pξqϕpξq ı “ 4 B Bξj ∆ξ ” pz ` LAq´1 ϕ ı ` ξj∆2 ξ ” pz ` LAq´1 pξqϕpξq ı (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='41) We have estimated the first term, so let us compute the second, ∆2 ξ ” pz ` LAq´1 ϕ ı “ ∆2 ξ pz ` LAq´1 ϕ ` 4∇ξ∆ξ pz ` LAq´1 ¨ ∇ξϕ ` 4 ” ∇2 ξ pz ` LAq´1 : ∇2 ξϕ ı ` 2∆ξ pz ` LAq´1 ∆ξϕ ` 4∇ξ pz ` LAq´1 ¨ ∇ξ∆ξϕ ` pz ` LAq´1 ∆2 ξϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='42) For the last four terms, we may estimate them by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24) again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the second term, since Bα ξ LA ≲ |ξ|1´|α|, by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='33), we may obtain ����� ż Bp1q eiθ¨ξξj∇ξ∆ξ pz ` LAq´1 pξq ¨ ∇ξϕ pξq dξ ����� ď ∥ϕ∥C1 4ÿ k“1 Cpkq δ,σ1,σ2,T 1 |z|k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='43) In the first term, we have ∆2 ξH “ ÿ j,k“1,2 r 8 pEjjkk ` Ejkjk ` Ejkkjq ´8 ` Ejkpjkq ` Ejpjkqk ` Epjkqjk ˘ ` 4H B2LA BξjBξk H B2LA BξjBξk H \uf6be ` ÿ j“1,2 “ ´4 ` Ejjpkkq ` Ejpkkqj ` Epkkqjj ˘ ` 4 ` Ejpjkkq ` Epjkkqj ˘‰ ` 2H∆ξLAH∆ξLAH ´ H∆2 ξLAH, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='44) where Ejjkk “H BLA Bξj H BLA Bξj H BLA Bξk H BLA Bξk H, Ejkpjkq “H BLA Bξj H BLA Bξk H B2LA BξjBξk H, Ejjpkkq “H BLA Bξj H BLA Bξj H∆ξLAH, Ejpjkkq “H BLA Bξk H B∆ξLA Bξj H, and we only have to compute the ´ pz ` LAq´1 ∆2 ξLA pz ` LAq´1 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since LA is an even function, ´ξj pz ` LAq´1 ∆2 ξLA pz ` LAq´1 pξq ϕpξq is odd, again, by WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 85 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='�����´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='Bp1q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='eiθ¨ξξj pz ` LAq´1 ∆2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ξLA pz ` LAq´1 pξq ϕpξqdξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='Bp1q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ξj sin pθ ¨ ξq pz ` LA pξqq´1 ∆2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ξLA pξq pz ` LA pξqq´1 ϕ pξq dξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='Bp1q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='θ ¨ ˆξ |ξ| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='¯ ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='z ` |ξ| LA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ˆξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='¯¯´1 ξjΦp4q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ˆξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='|ξ|3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='z ` |ξ| LA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ˆξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='¯¯´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ϕ p|ξ|q dξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='S1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ˆξj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ż 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='z ` rLA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ˆξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='¯¯´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='Φp4q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ˆξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='¯ ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='z ` rLA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ˆξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='¯¯´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ϕ prq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='θ ¨ ˆξr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='drdˆξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ď4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='S1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='���ˆξj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='���Φp4q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ˆξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='¯��� ∥ϕ prq∥C1pr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1sq ���� ´ z ` rLA ´ ˆξ ¯¯´1���� 2 C1pr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1sq dˆξ ďCδ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='T ∥ϕ prq∥C1pr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1sq ˜ 1 |z| ` 1 |z|2 ¸2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='45) Hence, ����� ż Bp1q eiθ¨ξξj pz ` LAq´1 pξqϕpξq ` eiθ¨ξ∆2 ξ ” ξj pz ` LAq´1 pξqϕpξq ı dξ ����� ď ∥ϕ prq∥C4 5ÿ k“1 Cpkq δ,σ1,σ2,T 1 |z|k ďCω,δ,σ1,σ2,T ∥ϕ∥C4 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='46) and ���� ż R2 eiθ¨ξξj pz ` LAq´1 pξqϕpξqdξ ���� ďCω,δ,σ1,σ2,T ∥ϕ∥C4 1 1 ` |θ|4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='47) □ Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given kpxq “ F´1re´LApξqs, then we have the following estimates ∥kpxq∥ ď C 1 1 ` |x|3 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='48) ���� B Bxi kpxq ���� ď C 1 1 ` |x|4 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='49) ���� B Bxi B Bxj kpxq ���� ď C 1 1 ` |x|5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='50) 86 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' First, p1 ` |x|3q ż R2 eix¨ξe´LApξqdξ “ ż R2p1 ´ i x |x| ¨ ix|x|2qeix¨ξe´LApξqdξ “ ż R2p1 ´ i x |x| ¨ ∇ξ|∇ξ|2qeix¨ξe´LApξqdξ “ ż R2 eix¨ξe´LApξq ` i x |x| ¨ ∇ξ∆ξe´LApξqdξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since e´LApξq ≲ e´|ξ|, we obtain (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='51) p1 ` |x|3q ∥kpxq∥ ≲ ´ 1 ` ����� ż B1p0q eix¨ξ x |x| ¨ ∇ξ∆ξe´LApξqdξ ����� ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' since B Bξi e´LApξq “ ´ ż 1 0 e´p1´tqLApξq B Bξi LApξqe´tLApξqdt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Bjkk ξ e´LApξq “ ´ ż 1 0 e´p1´t1qLApξqBjkk ξ LApξqe´t1LApξqdt1 ` H21 pj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' kkq ` H21 pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' jkq ` H21 pjk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' kq ` H22 pkk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' jq ` H22 pjk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' kq ` H22 pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' jkq ´ H3 pp1 ´ t1q p1 ´ t2q p1 ´ t3q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q p1 ´ t2q t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1q ´ H3 pp1 ´ t1q p1 ´ t2q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q t2 p1 ´ t3q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q t2t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1q ´ H3 pp1 ´ t1q p1 ´ t2q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1 p1 ´ t3q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1t3q ´ H3 pp1 ´ t1q p1 ´ t3q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1 p1 ´ t2q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1t2q ´ H3 pp1 ´ t1q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1 p1 ´ t2q p1 ´ t3q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1 p1 ´ t2q t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1t2q ´ H3 pp1 ´ t1q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1 p1 ´ t2q t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1t2 p1 ´ t3q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1t2t3q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' where H21 pα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' βq “ ż 1 0 ż 1 0 e´p1´t1qp1´t2qLApξqBα ξ LApξqe´p1´t1qt2LApξqBβ ξ LApξqe´t1LApξqdt1dt2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' H22 pα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' βq “ ż 1 0 ż 1 0 e´t1LApξqBα ξ LApξqe´p1´t1qt2LApξqBβ ξ LApξqe´p1´t1qp1´t2qLApξqdt1dt2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' H3 ps1, α, s2, β, s3, γ, s4q “ ż 1 0 ż 1 0 ż 1 0 e´s1LApξqBα ξ LApξqe´t2LApξqBβ ξ LApξqe´s3LApξqBγ ξ LApξqe´s4LApξqdt1dt2dt3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since LApξq is even and homogeneous of degree one, we have that the third deriva- tives of LApξq are odd and homogeneous of degree minus two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The other terms are less singular and thus the corresponding integrals in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='51) are bounded directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' That is to say, ���Bα ξ LApξq ��� ≲ |ξ|1´|ξ| and ��e´sLApξq�� ≲ e´sC|ξ| for some C ą 0, so ∥H21∥ , ∥H22∥ ≲ |ξ|´1 e´C|ξ|,∥H3∥ ≲ e´C|ξ|, and all of them are integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 87 Therefore, we only have to estimate the first, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ż B1p0q ż 1 0 e´p1´t1qLApξqeix¨ξ x |x| ¨ ∇ξ∆ξLApξqe´t1LApξqdt1dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We can write e´p1´t1qLApξq B Bξj ∆ξLApξqe´t1LApξq “ 1 |ξ|2 e´p1´t1qLApξqΦp3q A,jpˆξqe´t1LApξq, where ˆξ “ ξ{|ξ| and Φp3q A,jpˆξq is even and bounded from below and above, thanks to the arc-chord condition and the C1 regularity (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) and Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We then have that ż B1p0q ż 1 0 e´p1´t1qLApξqeix¨ξ x |x| ¨ ∇ξ∆ξLApξqe´t1LApξqdt1dξ “ ÿ j“1,2 ixj |x| ż S1 ż 1 0 ż 1 0 sin px ¨ ˆξ rq r e´p1´t1qrLApˆξqΦp3q A,jpˆξqe´t1rLApˆξqdrdt1dˆξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4, we obtain for all x P R2, ˆξ P S1 and t1 P r0, 1s ����� ż 1 0 sin px ¨ ˆξ rq r e´p1´t1qrLApˆξqΦp3q A,jpˆξqe´t1rLApˆξqdr ����� ď2 ���e´p1´t1qrLApˆξqΦp3q A,jpˆξqe´t1rLApˆξq��� C1pr0,1s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='rq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='52) LApˆξq is positive definite and diagonalizable and Φp3q A,jpˆξq is boundned, so for all ˆξ P S1 and t1 P r0, 1s, ���e´p1´t1qrLApˆξqΦp3q A,jpˆξqe´t1rLApˆξq��� C1pr0,1s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='rq ď C ´ 1 ` ���LApˆξq ��� ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='53) Therefore, since LApˆξq is bounded on S1, ż B1p0q ż 1 0 e´p1´t1qLApξqeix¨ξ x |x| ¨ ∇ξ∆ξLApξqe´t1LApξqdt1dξ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='54) is bounded, and we may conclude that ∥kpxq∥ ≲ p1 ` |x|3q´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, since B Bxi kpxq “ ż R2 iξie´LApξqeix¨ξdξ, we have ���� B Bxi kpxq ���� “ ż R2 |ξi| ���e´LApξq��� dξ ď C, where C only depends on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, |x|4 B Bxi kpxq “ ż R2 p∆ξq2 ´ iξie´LApξq¯ eix¨ξdξ, 88 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN where p∆ξq2 ` ξie´LApξq˘ “ 4 B Bξi ∆ξe´LApξq ` ξi p∆ξq2 e´LApξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The first term is the i component in ∇ξ∆ξe´LApξq, so we may claim the term is integratable by the previous techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the second term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Bjjkk ξ LApξq “ ´ ż 1 0 e´p1´t1qLApξqBjjkk ξ LApξqe´t1LApξqdt1 ` H2 pp1 ´ t1q p1 ´ t2q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' jjk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1q ` ¨ ¨ ¨ ´ H3 pp1 ´ t1q p1 ´ t2q p1 ´ t3q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' jj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q p1 ´ t2q t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1q ´ ¨ ¨ ¨ ` H4 pp1 ´ t1q p1 ´ t2q p1 ´ t3q p1 ´ t4q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q p1 ´ t2q p1 ´ t3q t4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q p1 ´ t2q t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1q ` ¨ ¨ ¨ where H2 ps1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' s3q “ ż 1 0 ż 1 0 e´s1LApξqBα ξ LApξqe´s2LApξqBβ ξ LApξqe´s3LApξqdt1dt2 H3 ps1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' s3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' s4q “ ż 1 0 ż 1 0 ż 1 0 e´s1LApξqBα ξ LApξqe´s2LApξqBβ ξ LApξqe´s3LApξqBγ ξ LApξqe´s4LApξq dt1dt2dt3 H4 ps1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' s3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' s4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' s5q “ ż 1 0 ż 1 0 ż 1 0 ż 1 0 e´s1LApξqBα ξ LApξqe´s2LApξqBβ ξ LApξqe´s3LApξqBγ ξ LApξqe´s4LApξq Bδ ξLApξqe´s5LApξqdt1dt2dt3dt4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' There are 14 H2-type terms, 36 H3-type terms, 24 H4-type terms in Bjjkk ξ LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since ���Bα ξ LApξq ��� ≲ |ξ|1´|ξ| and ��e´sLApξq�� ≲ e´sC|ξ| for some C ą 0, ∥ξiH2∥ ≲ |ξ|´1 e´C|ξ|,∥ξiH3∥ ≲ e´C|ξ| and ∥ξiH4∥ ≲ |ξ| e´C|ξ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, we only have to check ż R2 ż 1 0 e´p1´t1qLApξqξi p∆ξq2 LApξqe´t1LApξqdt1dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since ş1 0 e´p1´t1qLApξqξi p∆ξq2 LApξqe´t1LApξqdt1 is odd, we may use the same tech- nique in kpxq term to obtain |x|4 ���� B Bxi kpxq ���� ď C, so ���� B Bxi kpxq ���� ≲ 1 1 ` |x|4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, since B Bxi kpxq “ ż R2 ξie´LApξqeix¨ξdξ, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 89 we have ���� B Bxi kpxq ���� “ ż R2 |ξi| ���e´LApξq��� dξ ď C, where C only depends on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given a vector function M prq in C1 pr0, 1sq, we have the following inequality: for all A ě 0, ���� ż 1 0 M prq sin pArq r dr ���� ď 2 ∥Mp0q∥ ` ���� dM prq dr ���� C0pr0,1sq ď 2 ∥M∥C1pr0,1sq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='55) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ż 1 0 M prq sin pArq r dr “ ż 1 0 M p0q sin pArq r dr ` ż 1 0 M prq ´ M p0q r sin pArqdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='56) For the first term, ���� ż 1 0 M p0q sin pArq r dr ���� “ ����M p0q ż 1 0 sin pArq r dr ���� “ ∥Mp0q∥ ����� ż A 0 sin prq r dr ����� ď ∥Mp0q∥ ż π 0 sin prq r dr « 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='852 ∥Mp0q∥ ď 2 ∥Mp0q∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='57) For the second term, since ���� M prq ´ M p0q r ���� “ ��şr 0 dM ds psq ds �� r ď ���� dM prq dr ���� C0pr0,1sq , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='58) ���� ż 1 0 M prq ´ M p0q r sin pArqdr ���� ď ż 1 0 ���� M prq ´ M p0q r ���� |sin pArq| dr ď ���� dM prq dr ���� C0pr0,1sq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='59) Therefore, we obtain the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given fptq ě 0 is a locally integrable function on R, we have the following estimates: (i) If α ą ´1, for all m ă t, ���� ż t m pt ´ sqα fds ���� ď C pt ´ mqα`1 Mlrfsptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='60) (ii) If α ă ´1, for all M ă t, ����� ż M ´8 pt ´ sqα fds ����� ď C pt ´ Mqα`1 Mlrfsptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='61) 90 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (i) By integration by part theorem, ż t m pt ´ sqα fds “ ´ ż t m pt ´ sqα ˆ B Bs ż t s f prq dr ˙ ds “ pt ´ sqα ż t s f prq dr ˇˇˇˇ m s“t ´ α ż t m pt ´ sqα ˆ 1 t ´ s ż t s f prq dr ˙ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since lim sup sÑt´ ����pt ´ sqα ż t s f prq dr ���� “ lim sup sÑt´ ����pt ´ sqα`1 1 t ´ s ż t s f prq dr ���� ď lim sup sÑt´ pt ´ sqα`1 Mlrfsptq “ 0, �����pt ´ sqα ż t s f prq dr ˇˇˇˇ m s“t ����� ď pt ´ mqα`1 Mlrfsptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, ���� ż t m pt ´ sqα ˆ 1 t ´ s ż t s f prq dr ˙ ds ���� ďMlrfsptq ż t m pt ´ sqα ds “ 1 α ` 1 pt ´ mqα`1 Mlrfsptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, ���� ż t m pt ´ sqα fds ���� ď C pt ´ mqα`1 Mlrfsptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (ii) The proof is basically the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It will be ����� ż M ´8 pt ´ sqα fds ����� “ �����pt ´ sqα ż t s f prq dr ˇˇˇˇ M s“´8 ´ α ż M ´8 pt ´ sqα ˆ 1 t ´ s ż t s f prq dr ˙ ds ����� ď pt ´ Mqα`1 Mlrfsptq ` lim sup sÑ´8 pt ´ sqα`1 Mlrfsptq ` Mlrfsptq ż M ´8 pt ´ sqα ds “ pt ´ Mqα`1 Mlrfsptq ´ 1 α ` 1 pt ´ Mqα`1 Mlrfsptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' since α ` 1 ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, ����� ż M ´8 pt ´ sqα fds ����� ď C pt ´ Mqα`1 Mlrfsptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ References [1] Thomas Alazard and Omar Lazar, Paralinearization of the Muskat equation and appli- cation to the Cauchy problem, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 237 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2, 545–583, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/s00205-020-01514-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [2] Thomas Alazard and Quoc-Hung Nguyen, Endpoint sobolev theory for the Muskat equation, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 91 [3] , Quasilinearization of the 3D Muskat equation, and applications to the critical Cauchy problem, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 399 (2022), Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 108278, 52, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='aim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='108278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [4] David M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ambrose, Well-posedness of two-phase Hele-Shaw flow without surface tension, European J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 15 (2004), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 5, 597–607, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1017/S0956792504005662.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [5] , Well-posedness of two-phase Darcy flow in 3D, Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 65 (2007), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1, 189–203, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1090/S0033-569X-07-01055-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [6] David M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ambrose and Michael Siegel, Well-posedness of two-dimensional hydroe- lastic waves, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Edinburgh Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A 147 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3, 529–570, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1017/S0308210516000238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [7] Stephen Cameron, Global well-posedness for the two-dimensional Muskat problem with slope less than 1, Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' PDE 12 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 4, 997–1022, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2140/apde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [8] Stephen Cameron and Robert M Strain, Critical local well-posedness for the fully nonlinear Peskin problem, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='00692 (2021), Accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [9] Ke Chen and Quoc-Hung Nguyen, The Peskin problem with 9B1 8,8 initial data, arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13854 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [10] Ke Chen, Quoc-Hung Nguyen, and Yiran Xu, The Muskat problem with C1 data, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 375 (2021), 3039–3060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [11] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Arthur Cheng and Steve Shkoller, The interaction of the 3D Navier-Stokes equations with a moving nonlinear Koiter elastic shell, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 42 (2010), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3, 1094– 1155, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1137/080741628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [12] Peter Constantin, Diego C´ordoba, Francisco Gancedo, and Robert M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Strain, On the global existence for the Muskat problem, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (JEMS) 15 (2013), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1, 201–227, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4171/JEMS/360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [13] Antonio C´ordoba, Diego C´ordoba, and Francisco Gancedo, Interface evolution: the Hele-Shaw and Muskat problems, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (2) 173 (2011), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1, 477–542, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4007/annals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [14] Antonio C´ordoba, Diego C´ordoba, and Francisco Gancedo, Porous media: the Muskat prob- lem in three dimensions, Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' PDE 6 (2013), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2, 447–497, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2140/apde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [15] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Courant and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hilbert, Methods of mathematical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' I, Interscience Publishers, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', New York, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', 1953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [16] Daniel Coutand and Steve Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 179 (2006), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3, 303–352, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/s00205-005-0385-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [17] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Denisova, Problem of the motion of two viscous incompressible fluids separated by a closed free interface, Acta Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 37 (1994), 31–40, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/BF00995127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [18] Joachim Escher and Bogdan-Vasile Matioc, On the parabolicity of the Muskat problem: well- posedness, fingering, and stability results, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anwend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 30 (2011), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2, 193–218, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4171/ZAA/1431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [19] Evan A Evans and Richard Skalak, Mechanics and thermodynamics of biomembranes, CRC Press, 1980, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1201/9781351074339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [20] Thomas G Fai, Boyce E Griffith, Yoichiro Mori, and Charles S Peskin, Immersed boundary method for variable viscosity and variable density problems using fast constant-coefficient linear solvers i: Numerical method and results, SIAM Journal on Scientific Computing 35 (2013), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 5, B1132–B1161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [21] Gerald B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Folland, Introduction to partial differential equations, second ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', Princeton Uni- versity Press, Princeton, NJ, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [22] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Gancedo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Garc´ıa-Ju´arez, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Patel, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Strain, Global regularity for gravity unstable muskat bubbles, Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='02318 (2021), To appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [23] Francisco Gancedo, Rafael Granero-Belinch´on, and Stefano Scrobogna, Global existence in the lipschitz class for the N-peskin problem, Indiana Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='01787 (2021), To appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [24] Francisco Gancedo and Omar Lazar, Global well-posedness for the three dimensional Muskat problem in the critical Sobolev space, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 246 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1, 141–207, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/s00205-022-01808-x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [25] Eduardo Garc´ıa-Ju´arez, Yoichiro Mori, and Robert M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Strain, The Peskin Problem with Viscosity Contrast, Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' PDE, arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='03360 (2020), To appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 92 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN [26] Loukas Grafakos, Classical fourier analysis, third ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', Graduate Texts in Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 249, Springer, 2014, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/978-1-4939-1194-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [27] , Modern fourier analysis, third ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', Graduate Texts in Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 250, Springer, 2014, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/978-0-387-09434-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [28] Boyce E Griffith and Neelesh A Patankar, Immersed methods for fluid–structure interaction, Annual review of fluid mechanics 52 (2020), 421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [29] Susanna V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Haziot and Benoˆıt Pausader, Note on the dissipation for the general muskat problem, Preprint, arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='04395 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [30] Thomas Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hou, John S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lowengrub, and Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Shelley, Removing the stiffness from interfacial flows with surface tension, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 114 (1994), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2, 312–338, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1006/jcph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [31] William Ko and John M Stockie, Parametric resonance in spherical immersed elastic shells, SIAM Journal on Applied Mathematics 76 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1, 58–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [32] Daniel Lengeler and Michael Ruˇziˇcka, Weak solutions for an incompressible Newtonian fluid interacting with a Koiter type shell, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 211 (2014), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1, 205–255, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/s00205-013-0686-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [33] Hui Li, Stability of the Stokes immersed boundary problem with bending and stretching energy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 281 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 9, Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 109204, 65, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='jfa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='109204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [34] Fang-Hua Lin and Jiajun Tong, Solvability of the Stokes immersed boundary problem in two dimensions, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 72 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1, 159–226, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1002/cpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21764.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [35] Alessandra Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Mod- ern Birkh¨auser Classics, Birkh¨auser/Springer Basel AG, Basel, 1995, [2013 reprint of the 1995 original] [MR1329547].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [36] Bogdan-Vasile Matioc, The Muskat problem in two dimensions: equivalence of formu- lations, well-posedness, and regularity results, Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' PDE 12 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2, 281–332, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2140/apde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [37] Yoichiro Mori, Analise Rodenberg, and Daniel Spirn, Well-posedness and global behavior of the Peskin problem of an immersed elastic filament in Stokes flow, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 72 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 5, 887–980, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1002/cpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [38] Boris Muha and Suncica Cani´c, Existence of a weak solution to a nonlinear fluid- structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 207 (2013), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3, 919–968, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/s00205-012-0585-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [39] Huy Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Nguyen and Benoˆıt Pausader, A paradifferential approach for well-posedness of the Muskat problem, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 237 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1, 35–100, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/s00205-020-01494-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [40] Charles S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Peskin, Numerical analysis of blood flow in the heart, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 25 (1977), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3, 220–252, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1016/0021-9991(77)90100-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [41] , The immersed boundary method, Acta Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 11 (2002), 479–517, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1017/S0962492902000077.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [42] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Plotnikov and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Toland, Modelling nonlinear hydroelastic waves, Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 369 (2011), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1947, 2942–2956, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1098/rsta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='0104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [43] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Pozrikidis, Boundary integral and singularity methods for linearized viscous flow, Cam- bridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1992, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1017/CBO9780511624124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [44] , Modeling and simulation of capsules and biological cells, Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL, 2003, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1201/9780203503959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [45] Jan Pr¨uss and Gieri Simonett, Moving interfaces and quasilinear parabolic evolution equations, Monographs in Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 105, Birkh¨auser/Springer, [Cham], 2016, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/978-3-319-27698-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [46] Thomas Richter, Fluid-structure interactions: models, analysis and finite elements, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 118, Springer, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [47] Analise Rodenberg, 2D Peskin Problems of an Immersed Elastic Filament in Stokes Flow, ProQuest LLC, Ann Arbor, MI, 2018, Thesis (Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=')–University of Minnesota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [48] Yoshihiro Shibata, Theory of the lebesgue integral, Uchida Rokakuho, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 93 [49] Senjo Shimizu, Maximal regularity and its application to free boundary problems for the Navier-Stokes equations [translation of mr2656036], Sugaku Expositions 25 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1, 105–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [50] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Solonnikov, Solvability of the problem of the motion of a viscous incompressible fluid that is bounded by a free surface, Izv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Nauk SSSR Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 41 (1977), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 6, 1388–1424, 1448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [51] Jiajun Tong, Regularized Stokes immersed boundary problems in two dimensions: Well- posedness, singular limit, and error estimates, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 74(2):366–449 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [52] , Global solutions to the tangential Peskin problem in 2-D, Preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14723 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [53] Sunˇcica ˇCani´c, Marija Gali´c, and Boris Muha, Analysis of a 3D nonlinear, moving boundary problem describing fluid-mesh-shell interaction, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 373 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 9, 6621–6681, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1090/tran/8125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [54] Wei Wang, Pingwen Zhang, and Zhifei Zhang, Well-posedness of hydrodynamics on the moving elastic surface, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 206 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3, 953–995, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/s00205-012-0548-x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'}