Varying domains: Stability of the Dirichlet and the Poisson problem

Wolfgang Arendt and Daniel Daners
Preprint, February 2007
Discrete and Continuous Dynamical Systems - Series A, 21 (2008), 21 - 39.
Original article at doi:10.3934/dcds.2008.21.21
Citations on Google Scholar


For \(\Omega\) a bounded open set in \(\mathbb R^N\) we consider the space \(H^1_0(\bar{\Omega})=\{u_{|_{\Omega}} \colon u \in H^1(\mathbb R^N)\colon \text{\(u(x)=0\) 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 \quad\text{in \(\mathcal D(\Omega_n)^\prime\),} \qquad 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.

AMS Subject Classification (2000): 35J05, 31B05

A Preprint (PDF) is available.

You can go to the original article .