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 \[ \min\Big\{λ_1(Ω)+\dots+λ_k(Ω)\ :\ Ω\subset\mathbb{R}^d,\ \text{open}\ ,\ |Ω|=1\Big\}, \] where $λ_1(\cdot),\dots,λ_k(\cdot)$ denote the eigenvalues of the Dirichlet Laplacian and $|\cdot|$ the $d$-dimensional Lebesgue measure. We prove that the topological boundary of a minimizer $Ω_k^*$ is composed of a relatively open regular part which is locally a graph of a $C^{1,α}$ function and a closed singular part, which is empty 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.