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 gamma-convergence theory and the concentration- compactness principle. Given a sequence of open sets (A(n))(n is an element of N) in R-N, not necessarily bounded, but of uniformly bounded measure, w e prove a concentration-compactness result in L(L-2(RN)) for the sequence o f resolvent operators (R-An)(n is an element of N), where R-An : L-2(R-N) - -> H-0(1)(A(n)), RAn = (-Delta)(-1). (C) 2000 Academic Press. .