Regularity of the optimal sets for the second Dirichlet eigenvalue

TL;DR

This paper investigates the regularity of optimal sets minimizing a functional involving the second Dirichlet eigenvalue and volume, proving they are unions of regular disjoint open sets with well-characterized free boundaries.

Contribution

It establishes the regularity and geometric structure of optimal sets for the second Dirichlet eigenvalue problem, including their decomposition into two regular disjoint components.

Findings

Optimal sets are unions of two disjoint $C^{1,eta}$-regular open sets.

Free boundaries are smooth except on a set of Hausdorff dimension at most $d-5$.

The structure of minimizers is characterized in terms of one-phase free boundaries.

Abstract

This paper is dedicated to the regularity of the optimal sets for the second eigenvalue of the Dirichlet Laplacian. Precisely, we prove that if the set $\Omega$ minimizes the functional \[ \mathcal F_\Lambda(\Omega)=\lambda_2(\Omega)+\Lambda |\Omega|, \] among all subsets of a smooth bounded open set $D\subset \mathbb{R}^d$, where $\lambda_2(\Omega)$ is the second eigenvalue of the Dirichlet Laplacian on $\Omega$ and $\Lambda>0$ is a fixed constant, then $\Omega$ is equivalent to the union of two disjoint open sets $\Omega_+$ and $\Omega_-$, which are $C^{1,\alpha}$-regular up to a (possibly empty) closed set of Hausdorff dimension at most $d-5$, contained in the one-phase free boundaries $D\cap \partial\Omega_+\setminus\partial\Omega_-$ and $D\cap\partial\Omega_-\setminus\partial\Omega_+$.