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On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains (1711.07179v1)

Published 20 Nov 2017 in math.AP and math.NA

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 $\partial_{n} u=0$ 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 $p\in(1,\infty)$.

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