Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form

In this paper we consider minimizers of the functional$\min ?\{{\lambda }_1(\mathrm{\Omega })+?+{\lambda }_k(\mathrm{\Omega })+\mathrm{\Lambda }|\mathrm{\Omega }|,:\mathrm{\Omega }?D\text{ open}\}$ where $D?{\mathbb{R}}^d$ is a bounded open set and where $0<{\lambda }_1(\mathrm{\Omega })\le ?\le {\lambda }_k(\mathrm{\Omega })$ 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 ${\mathrm{\Omega }}^{?}$ have finite perimeter and that their free boundary $?{\mathrm{\Omega }}^{?}\cap D$ is composed of a regular part, which is locally the graph of a $C^{1,\alpha }$-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 $d^{?}\in \{5,6,7\}$.