Varying domains: Stability of the Dirichlet and the Poisson problem

Abstract

For $\mathrm{\Omega }$ a bounded open set in ${\mathbb{R}}^N$ we consider the space $H_0^1(\overline{\mathrm{\Omega }})=\{u_{{|}_{\mathrm{\Omega }}}:u\in H^1({\mathbb{R}}^N):u(x)=0\text{ a.e. outside }\overline{\mathrm{\Omega }}\}$. The set $\mathrm{\Omega }$ is called stable if $H_0^1(\mathrm{\Omega })=H_0^1(\overline{\mathrm{\Omega }})$. Stability of $\mathrm{\Omega }$ can be characterised by the convergence of the solutions of the Poisson equation $$-\mathrm{\Delta }{u}_n=f\text{in }\mathcal{D}({\mathrm{\Omega }}_n{)}^{\prime }\text{,}{u}_n\in H_0^1({\mathrm{\Omega }}_n)$$ and also the Dirichlet Problem with respect to ${\mathrm{\Omega }}_n$ if ${\mathrm{\Omega }}_n$ converges to $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ is supposed to be Dirichlet regular. However, stability does not imply Dirichlet regularity as Lebesgue's cusp shows.