*Accepted Paper*

**Inserted:** 5 sep 2016

**Last Updated:** 20 jan 2017

**Journal:** GAFA

**Year:** 2017

**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 \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.

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