This paper is dedicated to the spectral optimization problem min λ1(Ω) + · · · + λ k (Ω) + Λ|Ω| : Ω ⊂ D quasi-open 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.