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$\min \left\{{\lambda }_1\right(\Omega )+\cdots +{\lambda }_k(\Omega )+\Lambda |\Omega |,:\Omega \subset D\text{ open}\}$ where $D\subset R^d$ is a bounded open set and where $0<{\lambda }_1\left(\Omega \right)\le \cdots \le {\lambda }_k\left(\Omega \right)$ 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 ${\Omega }^{⁎}$ have finite perimeter and that their free boundary $\partial {\Omega }^{⁎}\cap D$ 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 $d^{⁎}\in \{5,6,7\}$.