On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains

TL;DR

This paper constructs a specific Lipschitz domain demonstrating that the known Sobolev regularity for solutions to Dirichlet and Neumann problems for the Laplacian cannot be improved beyond a certain limit, highlighting inherent regularity constraints.

Contribution

It provides explicit examples of Lipschitz domains where Sobolev regularity for Laplacian boundary value problems is exactly at the known limit, showing no further regularity gain is possible.

Findings

Existence of domains where regularity cannot be improved

Regularity limit is exactly $H^{3/2}$ for the Laplacian problems

Results extend to $L^{p}$ Sobolev spaces with $p eq 2$

Abstract

We construct a bounded $C^{1}$ domain $\Omega$ in $R^{n}$ for which the $H^{3/2}$ regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there exists $f$ in $C^{\infty}(\overline\Omega)$ such that the solution of $\Delta u=f$ in $\Omega$ and either $u=0$ on $\partial\Omega$ or $\partial\_{n} u=0$ on $\partial\Omega$ is contained in $H^{3/2}(\Omega)$ but not in $H^{3/2+\varepsilon}(\Omega)$ for any $\epsilon>0$. An analogous result holds for $L^{p}$ Sobolev spaces with $p\in(1,\infty)$.