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

Martin Costabel (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Martin Costabel. On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains. 2017. ⟨hal-01638088v1⟩