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Varying domains: Stability of the Dirichlet and the Poisson problem
1. | Institute of Applied Analysis, University of Ulm, D-89069 Ulm, Germany |
2. | School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia |
$ -\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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