# Varying domains: Stability of the Dirichlet and the Poisson problem

Preprint, February 2007
Discrete and Continuous Dynamical Systems - Series A, 21 (2008), 21 - 39.
Original article at doi:10.3934/dcds.2008.21.21
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.