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

TL;DR

This paper proves that small bi-Lipschitz domain deformations preserve the solvability of the $L^p$ Dirichlet problem for the Laplacian, unifying results for convex and $C^1$ domains.

Contribution

It introduces a novel change of variables based on the Green function to show stability of $L^p$ solvability under small bi-Lipschitz deformations.

Findings

Preservation of $L^p$ Dirichlet solvability under small bi-Lipschitz deformations.

Solvability results extend to small Lipschitz perturbations of convex domains.

Unification of convex and $C^1$ domain results for $L^p$ Dirichlet 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.