Existence and Regularity of Minimizers for Some Spectral Functionals with Perimeter Constraint

In this paper we prove that the shape optimization problem min {lambda(k)(Omega) : Omega subset of R-d, Omega open, P(Omega) = 1, vertical bar Omega vertical bar < +infinity} has a solution for any and dimension d. Moreover, every solution is a bounded connected open set with boundary which is C (1,alpha) outside a closed set of Hausdorff dimension d-8. Our results are more general and apply to spectral functionals of the form f(lambda(k)(Omega), ... ,lambda(kp)(Omega)) for increasing functions f satisfying some suitable bi-Lipschitz type condition.