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

摘要

In this paper we prove that the shape optimization problem min{λ_k(Ω): Ω ? ?~d, Ω open, P(Ω) = 1, |Ω| <+∞}, has a solution for any k ∈ N and dimension d. Moreover, every solution is a bounded connected open set with boundary which is C~(1,α) outside a closed set of Hausdorff dimension d - 8. Our results are more general and apply to spectral functionals of the form f (λ_(k1)(Ω),..., λ_(kp)(Ω)), for increasing functions f satisfying some suitable bi-Lipschitz type condition.