Calculus of Variations and Geometric Measure Theory

D. Mazzoleni - S. Terracini - B. Velichkov

Regularity of the optimal sets for some spectral functionals

created by mazzoleni on 05 Sep 2016
modified by velichkov on 21 Apr 2018


Published Paper

Inserted: 5 sep 2016
Last Updated: 21 apr 2018

Journal: Geom. Funct. Anal.
Year: 2017


In this paper we study the regularity of the optimal sets for the shape optimization problem \[ \min\Big\{\lambda_1(\Omega)+\dots+\lambda_k(\Omega)\ :\ \Omega\subset\mathbb{R}^d,\ \text{open}\ ,\
=1\Big\}, \] where $\lambda_1(\cdot),\dots,\lambda_k(\cdot)$ denote the eigenvalues of the Dirichlet Laplacian and $
$ the $d$-dimensional Lebesgue measure. We prove that the topological boundary of a minimizer $\Omega_k^*$ is composed of a relatively open \emph{regular part} which is locally a graph of a $C^{1,\alpha}$ function and a closed \emph{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.