This paper is dedicated to the spectral optimization problem \begin{equation*} \min \big\{ \lambda_1(\O)+\cdots+\lambda_k(\O) + \Lambda|\O| \ : \ \O \subset D \text{ quasi-open} \big\} \end{equation*}where D ⊂ ℝd is a bounded open set and 0 < λ1(Ω) ≤⋯ ≤ λk(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.