On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains - Archive ouverte HAL Access content directly
Journal Articles Mathematical News / Mathematische Nachrichten Year : 2019

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

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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Dates and versions

hal-01638088 , version 1 (19-11-2017)
hal-01638088 , version 2 (04-02-2019)

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Cite

Martin Costabel. On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains. Mathematical News / Mathematische Nachrichten, 2019, 292 (10), pp.2165-2173. ⟨10.1002/mana.201800077⟩. ⟨hal-01638088v2⟩
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