Varying domains: Stability of the Dirichlet and the Poisson problem

Wolfgang Arendt; Daniel Daners

doi:10.3934/dcds.2008.21.21

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2008, Volume 21, Issue 1: 21-39. Doi: 10.3934/dcds.2008.21.21

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Varying domains: Stability of the Dirichlet and the Poisson problem

Wolfgang Arendt^1, and
Daniel Daners^2,

1.
Institute of Applied Analysis, University of Ulm, D-89069 Ulm
2.
School of Mathematics and Statistics, The University of Sydney, NSW 2006

Received: February 28, 2007

Revised: September 30, 2007

Published: December 2007

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Abstract

For $\Omega$ a bounded open set in $\R^N$ we consider the space $H^1_0(\bar{\Omega})=${$u_{|_{\Omega}}: u \in H^1(\R^N):$ $u(x)=0$ a.e. outside $\bar{\Omega}$}. The set $\Omega$ is called stable if $H^1_0(\Omega)=H^1_0(\bar{\Omega})$. Stability of $\Omega$ can be characterised by the convergence of the solutions of the Poisson equation

$ -\Delta u_n = f$ in $D(\Omega_n)^$´, $ u_n \in H^1_0(\Omega_n)$

and also the Dirichlet Problem with respect to $\Omega_n$ if $\Omega_n$ converges to $\Omega$ in a sense to be made precise. We give diverse results in this direction, all with purely analytical tools not referring to abstract potential theory as in Hedberg's survey article [Expo. Math. 11 (1993), 193--259]. The most complete picture is obtained when $\Omega$ is supposed to be Dirichlet regular. However, stability does not imply Dirichlet regularity as Lebesgue's cusp shows.

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Mathematics Subject Classification: Primary: 35J05, Secondary: 31B05.

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