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

Abstract

In this paper we consider minimizers of the functional \begin{equation*} \min \big\{ \lambda_1(\Omega)+\cdots+\lambda_k(\Omega) + \Lambda|\Omega|, \ : \ \Omega \subset D \text{ open} \big\} \end{equation*} where $D\subset\mathbb{R}^d$ is a bounded open set and where $0<\lambda_1(\Omega)\leq\cdots\leq\lambda_k(\Omega)$ are the first $k$ eigenvalues on $\Omega$ of an operator in divergence form with Dirichlet boundary condition and with H\"{o}lder continuous coefficients. We prove that the optimal sets $\Omega^\ast$ have finite perimeter and that their free boundary $\partial\Omega^\ast\cap D$ 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 $d<d^\ast$, discrete if $d=d^\ast$ and of Hausdorff dimension at most $d-d^\ast$ if $d>d^\ast$, for some $d^\ast\in\{5,6,7\}$.