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\{ λ_1(Ω)+\cdots+λ_k(Ω) + Λ|Ω| \ : \ Ω\subset D \text{ quasi-open} \big\} \end{equation*} where $D\subset {\mathbb{R}}^d$ is a bounded open set and $0<{\lambda }_1(\Omega )\le \cdots \le {\lambda }_k(\Omega )$ are the first $k$ eigenvalues on $\Omega$ of an operator in divergence form with Dirichlet boundary condition and Hölder 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.