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

This paper is dedicated to the spectral optimization problem min{λ1 (Ω)+⋯+λk (Ω)+Λ|Ω| : Ω⊂D quasi-open } \begin{equation*} \min \big\{ \lambda_1(\O)+\cdots+\lambda_k(\O) + \Lambda|\O| \ : \ \O \subset D \text{ quasi-open} \big\} \end{equation*} minλ1(Ω)+⋯+λk(Ω)+Λ|Ω| :Ω⊂D quasi-open where D ⊂ ℝd is a bounded open set and 0 1 (Ω) ≤⋯ ≤ λ k (Ω) are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and Holder 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.