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

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 $\partial \_nu=0$ on $\partial \Omega$ is contained in $H^{3/2}(\Omega )$ but not in $H^{3/2+\epsilon }(\Omega )$ for any $\epsilon >0$. An analogous result holds for $L^p$ Sobolev spaces with $p\in (1,\infty )$.