Varying domains: Stability of the Dirichlet and the Poisson problem
Wolfgang Arendt  Institute of Applied Analysis, University of Ulm, D89069 Ulm, Germany (email) 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 $D(\Omega_n)^$ยด, $ 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), 193259]. 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.
Keywords: stability,
Poisson problem, Dirichlet problem, harmonic function.
Received: March 2007; Revised: October 2007; Available Online: February 2008. 
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