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Stability of $L^p$ Dirichlet solvability under small bi-Lipschitz transformations of domains

Joseph Feneuil, Linhan Li, Jinping Zhuge

TL;DR

This work proves that the solvability of the L^p Dirichlet problem for the Laplacian on a Lipschitz domain is stable under small bi-Lipschitz deformations, preserving the same p in (1,∞). The authors introduce a novel Green-function–based change of variables that encodes perturbations in a non-flat base domain, and they construct an intermediate operator whose Carleson-perturbation controls transfer of solvability to the perturbed domain. By combining Carleson perturbation stability with bi-Lipschitz invariance, they establish (D)_p for all p>1 on small perturbations of convex domains and, more generally, on strongly quasiconvex domains. The results unify convex and C^1-type settings and extend stability to perturbations not limited to the transversal direction, with potential applicability to broader elliptic operators and Neumann problems.

Abstract

We show that small bi-Lipschitz deformations of a Lipschitz domain (with possibly large Lipschitz constant) preserve the solvability of the Dirichlet problem for the Laplacian with boundary data in $L^p$, for the same value of $p>1$. As a consequence, for all $p\in(1,\infty)$, we obtain the solvability of the $L^p$ Dirichlet problem for small Lipschitz perturbations of convex domains, thereby unifying two fundamentally different settings in which such results were previously known: convex and $C^1$ domains. The key ingredient and novelty of our approach is a construction of a change of variables based on a non-constant basis derived from the Green function, which encodes the geometry of the base domain.

Stability of $L^p$ Dirichlet solvability under small bi-Lipschitz transformations of domains

TL;DR

This work proves that the solvability of the L^p Dirichlet problem for the Laplacian on a Lipschitz domain is stable under small bi-Lipschitz deformations, preserving the same p in (1,∞). The authors introduce a novel Green-function–based change of variables that encodes perturbations in a non-flat base domain, and they construct an intermediate operator whose Carleson-perturbation controls transfer of solvability to the perturbed domain. By combining Carleson perturbation stability with bi-Lipschitz invariance, they establish (D)_p for all p>1 on small perturbations of convex domains and, more generally, on strongly quasiconvex domains. The results unify convex and C^1-type settings and extend stability to perturbations not limited to the transversal direction, with potential applicability to broader elliptic operators and Neumann problems.

Abstract

We show that small bi-Lipschitz deformations of a Lipschitz domain (with possibly large Lipschitz constant) preserve the solvability of the Dirichlet problem for the Laplacian with boundary data in , for the same value of . As a consequence, for all , we obtain the solvability of the Dirichlet problem for small Lipschitz perturbations of convex domains, thereby unifying two fundamentally different settings in which such results were previously known: convex and domains. The key ingredient and novelty of our approach is a construction of a change of variables based on a non-constant basis derived from the Green function, which encodes the geometry of the base domain.
Paper Structure (17 sections, 23 theorems, 217 equations)

This paper contains 17 sections, 23 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. Motivation and statement of main results
  3. Existing methods and their limitations
  4. Main ideas of our approach
  5. Preliminaries
  6. Notations and definitions
  7. Some properties on $(D)_p$
  8. Estimates for the Green function.
  9. The change of variables - the heart of the matter
  10. Set-up estimates on Lipschitz graph domains
  11. Quantifying small Lipschitz perturbations of Lipschitz graph domains.
  12. Construction of the change of variables
  13. Proof of the main theorem and corollary
  14. Proof of Theorem \ref{['MainTh']}
  15. Proof of Corollary \ref{['MainCor']}
  16. ...and 2 more sections

Key Result

Theorem 1.3

Let $\Omega_0\subset\mathbb R^n$ be a Lipschitz graph domain, that is, there exists $M>0$ and a $M$-Lipschitz function $g:\mathbb R^{n-1}$ such that Let $p\in (1,\infty)$ be such that the $L^p$ Dirichlet problem for the Laplacian is solvable in $\Omega_0$. There exists $\epsilon_0>0$ such that if $\Phi :\, \mathbb R^n \to \mathbb R^n$ is a bi-Lipschitz map whose Jacobian matrix $\nabla\Phi$ satis

Theorems & Definitions (40)

  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.7
  • Definition 1.8
  • Remark 1.9
  • Corollary 1.10
  • Corollary 1.11
  • Definition 2.1: $\mathop{\mathrm{CM_{sup}}}\nolimits$ and $\mathop{\mathrm{CM}}\nolimits$
  • Remark 2.2
  • ...and 30 more