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

Abstract

We construct a bounded $C^{1}$ domain $Ω$ 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 $C^{\infty}(\overlineΩ)$ such that the solution of $Δu=f$ in $Ω$ and either $u=0$ on $\partialΩ$ or $\partial\_{n} u=0$ on $\partialΩ$ is contained in $H^{3/2}(Ω)$ but not in $H^{3/2+\varepsilon}(Ω)$ for any $ε>0$. An analogous result holds for $L^{p}$ Sobolev spaces with $p\in(1,\infty)$.