Regularity of optimal sets for spectral functionals

We consider the variational shape optimization problem of the minimization of the sum of the first $k$ Dirichlet eigenvalues of a set $\Omega$ under a volume constraint $|\Omega|=1$. We prove that the free boundary of the optimal set is $C^{1,\alpha}$ regular up to a set of zero (d-1)-Hausdorff measure. The optimal set is a solution of a free boundary problem of Alt-Caffarelli type involving vector valued functions. We will dedicate most of our attention to the study of the local minimizers for this free boundary problem.