## Abstract

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.