Uniform concentration-compactness for Sobolev spaces on variable domains

Abstract

We present a new method for proving existence results in shape optimization problems involving the eigenvalues of the Dirichlet-Laplace operator. This method brings together the γ-convergence theory and the concentration-compactness principle. Given a sequence of open sets (An)n∈N in RN, not necessarily bounded, but of uniformly bounded measure, we prove a concentration-compactness result in L(L2(RN)) for the sequence of resolvent operators (RAn)n∈N, where RAn:L2(RN)→H10(An), RAn=(-Δ)-1. © 2000 Academic Press.