Lipschitz continuity of the eigenfunctions on optimal sets for functionals with variable coefficients

Abstract

This paper is dedicated to the spectral optimization problem \begin{equation*} \min \big\{ \lambda_1(\Omega)+\cdots+\lambda_k(\Omega) + \Lambda|\Omega| \ : \ \Omega \subset D \text{ quasi-open} \big\} \end{equation*} where $D\subset\mathbb{R}^d$ is a bounded open set and $0<\lambda_1(\Omega)\leq\cdots\leq\lambda_k(\Omega)$ are the first $k$ eigenvalues on $\Omega$ of an operator in divergence form with Dirichlet boundary condition and H\"{o}lder continuous coefficients. We prove that the first $k$ eigenfunctions on an optimal set for this problem are locally Lipschtiz continuous in $D$ and, as a consequence, that the optimal sets are open sets. We also prove the Lipschitz continuity of vector-valued functions that are almost-minimizers of a two-phase functional with variable coefficients.