Varying domains: Stability of the Dirichlet and the Poisson problem

Abstract

For Ω a bounded open set in R^N we consider the space H₀¹() = {u |_Ω : u ∈ H¹(R^N), u(x) = 0 a.e. outside }. The set Ω is called stable if H₀¹(Ω) = H ₀¹(). Stability of Ω can be characterised by the convergence of the solutions of the Poisson equation

and also the Dirichlet Problem with respect to Ω_n if Ω_n converges to Ω 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 Ω is supposed to be Dirichlet regular. However, stability does not imply Dirichlet regularity as Lebesgue’s cusp shows.