Uniform Concentration-Compactness for Sobolev Spaces on Variable Domains

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 in N, not necessarily bounded, but of uniformly bounded measure, we prove a concentration-compactness result in (L2( N)) for the sequence of resolvent operators (RAn)n , where RAn: L2( N)→H10(An), RAn=(−Δ)−1.