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 $p\in(1,\infty)$.
Categories: math.AP math.NA