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 $\Lambda>0, D\subset \mathbb{R}^n$ a bounded open set and $\lambda_i^s(\Omega)$ $i$-th eigenvalues of fractional Laplacian on $\Omega$ with Dirichlet boundary condition $\mathbb{R}^n\setminus \Omega$. We first prove that $m$ eigenfunctions an optimal are locally H\"{o}lder continuous in class $C^{0,s}$ and, as consequence, sets sets. Then, via blow-up analysis based Weiss type monotonicity formula, we topological minimizer composed relatively regular part closed singular Hausdorff dimension at most $n-n^*$, for some $n^*\geq 3$. Finally use viscosity approach $C^{1,\alpha}$-regularity boundary.