AN EXISTENCE RESULT FOR A CLASS OF SHAPE OPTIMIZATION PROBLEMS

Given a bounded open subset OMEGA of R(n), we prove the existence of a minimum point for a functional F defined on the family A(OMEGA) of all ''quasi-open'' subsets of OMEGA, under the assumption that F is decreasing with respect to set inclusion and that F is lower semicontinuous on A(OMEGA) 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.