Varying domains: Stability of the Dirichlet and the Poisson problem (Q939254)

For \(\Omega\) being a bounded open set in \(\mathbb{R}^N\) the authors consider the space \(H_0^1(\overline{\Omega}) = \{u| _{\Omega} : u\in H^1(\mathbb{R}^N):u(x)=0\) a.e. outside \(\overline{\Omega}\}\). The set \(\Omega\) is called stable if \(H_0^1(\Omega) = H_0^1(\overline{\Omega})\). Stability of \(\Omega\) can be characterized by the convergence of the solutions of the Dirichlet problem for the Poisson equation \(-\Delta u_n =f\) in \(\mathcal{D}(\Omega_n)'\) (\(u_n \in H_0^1(\Omega)\)) with respect to \(\Omega_n \to \Omega\). 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.