Regularity of shape optimizers for some spectral fractional problems

This paper is dedicated to the spectral optimization problem $$\mathrm{min}\left\{\lambda_1^s(\Omega)+\cdots+\lambda_m^s(\Omega) + \Lambda \mathcal{L}_n(\Omega)\colon \Omega\subset D \mbox{ s-quasi-open}\right\}$$ where $\Lambda>0, D\subset \mathbb{R}^n$ 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 $\mathbb{R}^n\setminus \Omega$. 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.