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

Abstract

In this paper we prove that the shape optimization problem $$\min \left\{{\lambda }_k(\Omega ):\Omega \subset {\mathbb{R}}^d,\Omega open,P(\Omega )=1,|\Omega |<+\infty \right\},$$ has a solution for any $k\in \mathbb{N}$ 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 $f({\lambda }_{k_1}(\Omega ),\dots ,{\lambda }_{k_p}(\Omega ))$, for increasing functions $f$ satisfying some suitable bi-Lipschitz type condition.