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 , 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.
Keywords
Cite
@article{arxiv.2602.08115,
title = {Stability of $L^p$ Dirichlet problem under small bi-Lipschitz transformations of domains},
author = {Joseph Feneuil and Linhan Li and Jinping Zhuge},
journal= {arXiv preprint arXiv:2602.08115},
year = {2026}
}
Comments
42 pages. Added further details to the proof of Theorem 3.45 to show the global invertibility of the map $\rho$. 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