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 ${\lambda }_1(\cdot ),\dots ,{\lambda }_k(\cdot )$ denote the eigenvalues of the Dirichlet Laplacian and $|\cdot |$ the $d$-dimensional Lebesgue measure. We prove that the topological boundary of a minimizer ${\Omega }_k^{\ast }$ is composed of a relatively open regular part which is locally a graph of a $C^{1,\alpha }$ function and a closed singular part, which is empty if $d<d^{\ast }$, contains at most a finite number of isolated points if $d=d^{\ast }$ and has Hausdorff dimension smaller than $(d-d^{\ast })$ if $d>d^{\ast }$, where the natural number $d^{\ast }\in [5,7]$ 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.

arXiv comment32 pages, New version: a proof of the $C^{\infty }$ regularity (in the previous version it was only $C^{1,\alpha }$) of the regular part of the free boundary was added. Few typos and misprints corrected