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

TL;DR

This paper proves that eigenfunctions on optimal sets for a spectral optimization problem with variable coefficients are locally Lipschitz continuous, leading to the conclusion that these optimal sets are open, and extends Lipschitz regularity to almost-minimizers of two-phase functionals.

Contribution

It establishes Lipschitz continuity of eigenfunctions on optimal sets with variable coefficients and shows these sets are open, advancing spectral optimization theory.

Findings

Eigenfunctions are locally Lipschitz continuous on optimal sets.

Optimal sets are proven to be open.

Lipschitz continuity extends to almost-minimizers of two-phase functionals.

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.