Regularity of shape optimizers for some spectral fractional problems

This paper is dedicated to the spectral optimization problem $$ mathrm{min}left{lambda_1^s(Omega)+cdots+lambda_m^s(Omega) + Lambda mathcal{L}_n(Omega)colon Omegasubset D mbox{ s-quasi-open}right} $$ where $Lambda>0, Dsubset mathbb{R}^n$ is a bounded open set and $lambda_i^s(Omega)$ is the $i$-th eigenvalues of the fractional Laplacian on $Omega$ with Dirichlet boundary condition on $mathbb{R}^nsetminus Omega$. We first prove that the first $m$ eigenfunctions on an optimal set are locally H{o}lder continuous in the class $C^{0,s}$ and, as a consequence, that the optimal sets are open sets. Then, via a blow-up analysis based on a Weiss type monotonicity formula, we prove that the topological boundary of a minimizer $Omega$ is composed of a relatively open regular part and a closed singular part of Hausdorff dimension at most $n-n^*$, for some $n^*geq 3$. Finally we use a viscosity approach to prove $C^{1,alpha}$-regularity of the regular part of the boundary.