Domain perturbations and estimates for the solutions of second order elliptic equations

Abstract

We study the dependence of the variational solution of the inhomogeneous Dirichlet problem for a second order elliptic equation with respect to perturbations of the domain. We prove optimal L 2 and energy estimates for the difference of two solutions in two open sets in terms of the “distance” between them and suitable geometrical parameters which are related to the regularity of their boundaries. We derive such estimates when at least one of the involved sets is uniformly Lipschitz: due to the connection of this problem with the regularity properties of the solutions in the L 2 family of Sobolev–Besov spaces, the Lipschitz class is the reasonably weakest one compatible with the optimal estimates.