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

Abstract

In this paper we prove that the shape optimization problem $$\min \bigl\{\lambda_k(\varOmega):\ \varOmega\subset \mathbb{R}^d,\ \varOmega\ \hbox{open},\ P(\varOmega)=1,\ |\varOmega|<+\infty \bigr\}, $$ 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,α outside a closed set of Hausdorff dimension d−8. Our results are more general and apply to spectral functionals of the form $f(\lambda_{k_{1}}(\varOmega),\dots,\lambda_{k_{p}}(\varOmega))$ , for increasing functions f satisfying some suitable bi-Lipschitz type condition.