Regularity of shape optimizers for some spectral fractional problems

Abstract

This paper is dedicated to the spectral optimization problem $$ \mathrm{min}\left\{λ_1^s(Ω)+\cdots+λ_m^s(Ω) + Λ\mathcal{L}_n(Ω)\colon Ω\subset D \mbox{ s-quasi-open}\right\} $$ 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 Hölder 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 $n-n^{\ast }$, for some $n^{\ast }\ge 3$. Finally we use a viscosity approach to prove $C^{1,\alpha }$-regularity of the regular part of the boundary.

arXiv admin note: text overlap with arXiv:2010.05782