An existence result for a class of shape optimization problems

Given a bounded open subset of R ⁿ, we prove the existence of a minimum point for a functional F defined on the family A() of all quasiopen subsets of , under the assumption that F is decreasing with respect to set inclusion and that F is lower semicontinuous on A() with respect to a suitable topology, related to the resolvents of the Laplace operator with Dirichlet boundary condition. Applications are given to the existence of sets of prescribed volume with minimal k ^th eigenvalue (or with minimal capacity) with respect to a given elliptic operator.