Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Varying domains: Stability of the Dirichlet and the Poisson problem

Pages: 21 - 39, Volume 21, Issue 1, May 2008      doi:10.3934/dcds.2008.21.21

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Wolfgang Arendt - Institute of Applied Analysis, University of Ulm, D-89069 Ulm, Germany (email)
Daniel Daners - School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia (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), 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.

Keywords:  stability, Poisson problem, Dirichlet problem, harmonic function.
Mathematics Subject Classification:  Primary: 35J05, Secondary: 31B05.

Received: March 2007;      Revised: October 2007;      Available Online: February 2008.