Regularity of the optimal sets for some spectral functionals

TL;DR

This paper investigates the regularity of optimal sets in a spectral shape optimization problem involving the sum of the first k eigenvalues of the Dirichlet Laplacian, revealing detailed boundary structure and extending free boundary regularity theory.

Contribution

It provides the first extension of free boundary regularity theory to vector-valued cases and characterizes the boundary regularity of spectral optimization minimizers.

Findings

Boundary has a regular part that is a $C^{1,eta}$ graph.

Singular part is empty for dimensions below $d^*$.

Singularities are finite or of lower Hausdorff dimension depending on dimension.

Abstract

In this paper we study the regularity of the optimal sets for the shape optimization problem \[ \min\Big\{\lambda_1(\Omega)+\dots+\lambda_k(\Omega)\ :\ \Omega\subset\mathbb{R}^d,\ \text{open}\ ,\ |\Omega|=1\Big\}, \] 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 $\Omega_k^*$ 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 $d<d^*$, 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.