In this paper we consider minimizers of the functional min λ1(Ω) + · · · + λ k (Ω) + Λ|Ω|, : Ω ⊂ D open where D ⊂ R d is a bounded open set and where 0 < λ1(Ω) ≤ · · · ≤ λ k (Ω) are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets Ω * have finite perimeter and that their free boundary ∂Ω * ∩ D is composed of a regular part, which is locally the graph of a C 1,α-regular function, and a singular part, which is empty if d < d * , discrete if d = d * and of Hausdorff dimension at most d − d * if d > d * , for some d * ∈ {5, 6, 7}.