Lipschitz Regularity of the Eigenfunctions on Optimal Domains
Résumé
We study the optimal sets $\Omega^*\subseteq\mathbb{R}^d$ for spectral functionals of the form $F(\lambda_1(\Omega),\dots,\lambda_p(\Omega))$, which are bi-Lipschitz with respect to each of the eigenvalues $\lambda_1(\Omega),\lambda_2(\Omega),\dots,\lambda_p(\Omega))$ of the Dirichlet Laplacian on $\Omega$, a prototype being the problem
$$\min\{\lambda_1(\Omega)+\dots+\lambda_p(\Omega): \Omega\subseteq\mathbb{R}^d, |\Omega|=1\}$$
We prove the Lipschitz regularity of the eigenfunctions $u_1,\dots,u_p$ of the Dirichlet Laplacian on the optimal set $\Omega^*$ and, as a corollary, we deduce that $\Omega^*$ 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.