Regularity of the optimal sets for some spectral functionals

Abstract

In this paper we study the regularity of the optimal sets for the shape optimization problem $$\min\Big\{\lambda_{1}(\Omega)+\dots+\lambda_{k}(\Omega) : \Omega \subset {\mathbb {R}}^{d} {\rm open},\quad |\Omega| = 1\Big\},$$ where $${\lambda_{1}(\cdot),\ldots,\lambda_{k}(\cdot)}$$ denote the eigenvalues of the Dirichlet Laplacian and $${|\cdot|}$$ the d-dimensional Lebesgue measure. We prove that the topological boundary of a minimizer $${\Omega_{k}^{*}}$$ is composed of a relatively open regular part which is locally a graph of a $${C^{\infty}}$$ function and a closed singular part, which is empty if $${d < d^{*}}$$ , contains at most a finite number of isolated points if $${d = d^{*}}$$ and has Hausdorff dimension smaller than $${(d-d^{*})}$$ if $${d > d^{*}}$$ , where the natural number $${d^{*} \in [5,7]}$$ is the smallest dimension at which minimizing one-phase free boundaries admit singularities. To achieve our goal, as an auxiliary result, we shall extend for the first time the known regularity theory for the one-phase free boundary problem to the vector-valued case.