VARYING DOMAINS: STABILITY OF THE DIRICHLET AND THE POISSON PROBLEM

摘要: For $\Omega$ a bounded open set in $\R^N$ we consider the space $H^1_0(\bar{\Omega})=${$u_{|_{\Omega}}: u \in H^1(\R^N):$ $u(x)=0$ a.e. outside $\bar{\Omega}$}. The is called stable if $H^1_0(\Omega)=H^1_0(\bar{\Omega})$. Stability of can be characterised by convergence the solutions Poisson equation $ -\Delta u_n = f$ $D(\Omega_n)^$´, H^1_0(\Omega_n)$ and also Dirichlet Problem with respect to $\Omega_n$ if converges sense made precise. We give diverse results this direction, all purely analytical tools not referring abstract potential theory as Hedberg's survey article [Expo. Math. 11 (1993), 193--259]. most complete picture is obtained when supposed Dirichlet regular. However, stability does imply regularity as Lebesgue's cusp shows.