Regularity of shape optimizers for some spectral fractional problems

TL;DR

This paper investigates the regularity and structure of optimal shapes in a spectral fractional problem, proving local Hölder continuity of eigenfunctions, openness of optimal sets, and boundary regularity using advanced analytical techniques.

Contribution

It establishes the regularity of eigenfunctions and the boundary of optimal sets in fractional spectral optimization, introducing new methods for analyzing their geometric properties.

Findings

Eigenfunctions are locally Hölder continuous in $C^{0,s}$.

Optimal sets are proven to be open.

The boundary has a regular part and a singular part with controlled Hausdorff dimension.

Abstract

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.