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

Université Grenoble Alpes, CNRS UMR 5582, Institut Fourier, 38610 Gières, France.

^* Corresponding author: baptiste.trey@univ-grenoble-alpes.fr

Received: 4 October 2019
Accepted: 9 March 2020

Abstract

This paper is dedicated to the spectral optimization problem

where D ⊂ ℝ^d is a bounded open set and 0 < λ₁(Ω) ≤⋯ ≤ λ_k(Ω) are the first k eigenvalues on Ω 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.