Existence and regularity of minimizers for some spectral functionals with perimeter constraint

In this paper we prove that the shape optimization problem {λk (Ω) : Ω ⊂ ℝd, 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 λk1 (Ω)⋯ λkp (Ω)), for increasing functions f satisfying some suitable bi-Lipschitz type condition. © 2013 Springer Science+Business Media New York.