Lipschitz Regularity of the Eigenfunctions on Optimal Domains

Abstract

We study the optimal sets ${\Omega^\ast\subseteq\mathbb{R}^d}$ for spectral functionals of the form ${F\big(\lambda_1(\Omega),\ldots,\lambda_p(\Omega)\big)}$, which are bi-Lipschitz with respect to each of the eigenvalues ${\lambda_1(\Omega), \lambda_2(\Omega)}, \ldots, {\lambda_p(\Omega)}$ of the Dirichlet Laplacian on ${\Omega}$, a prototype being the problem

$$\min{\big\{\lambda_1(\Omega)+\cdots+\lambda_p(\Omega)\;:\;\Omega\subseteq\mathbb{R}^d,\ |\Omega|=1\big\}}.$$

We prove the Lipschitz regularity of the eigenfunctions ${u_1,\ldots,u_p}$ of the Dirichlet Laplacian on the optimal set ${\Omega^\ast}$ and, as a corollary, we deduce that ${\Omega^\ast}$ is open. For functionals depending only on a generic subset of the spectrum, as for example ${\lambda_k(\Omega)}$, our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved.