  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
$p\in(1,\infty)$.
