Uniform concentration-compactness for Sobolev spaces on variable domains (Q1566847)

The author presents a new method for proving existence results for some classes of shape optimization problems involving the eigenvalues of Laplace operator with Dirichlet homogeneous boundary conditions. Such method exploits the concentration-compactness principle, together with the \(\gamma\)-convergence theory of sets. The main result of the paper is expressed as a concentration-compactness theorem in \({\mathcal L}(L^2({\mathbb R}^N))\) for the sequence of the resolvent operators \(\{R_{A_n}\}_{n\in{\mathbb N}}\) of the Laplace operator with Dirichlet homogeneous boundary conditions, where \(\{A_n\}_{n\in{\mathbb N}}\) is a sequence of open (or quasi open) sets with uniformly bounded Lebesgue measure. The analysis of the asymptotic behaviour of \(\{A_n\}_{n\in{\mathbb N}}\), in the sense of \(\gamma\)-convergence, and of \(\{R_{A_n}\}_{n\in{\mathbb N}}\), in the sense of the uniform operator topology of \(L^2({\mathbb R}^N)\), is carried out, and the various possible situations are described. The results are then applied for proving existence results of global minimizers in sets of quasi open sets, not necessarily bounded by a uniform bound, but just with uniformly bounded measures, for a class of shape optimization problems depending on the eigenvalues of Laplace operator with Dirichlet homogeneous boundary conditions.