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

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.

42 pages. Added further details to the proof of Theorem 3.45 to show the global invertibility of the map $\ddot{I}$. Expanded part of the proof of Theorem 3.23 into a separate lemma (Lemma 3.14). Moved the proof of Lemma 3.2 to the Appendix as an alternative proof