Varying domains: Stability of the dirichlet and the poisson problem

Abstract

For Ω a bounded open set in ℝN we consider the space H01(Ω̄) = {u|Ω : u ∈ H1(ℝN): u(x) = 0 a.e. outside Ω̄}. The set Ω is called stable if H01(Ω) = H 01(Ω̄). Stability of Ω can be characterised by the convergence of the solutions of the Poisson equation -Δun = f in D(Ωn)′, un ∈ H01(Ωn) and also the Dirichlet Problem with respect to Ωn if Ωn 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.