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

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