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\{ λ_1(Ω)+\cdots+λ_k(Ω) + Λ|Ω|, \ : \ Ω\subset D \text{ open} \big\} \end{equation*} where $D\subset {\mathbb{R}}^d$ is a bounded open set and where $0<{\lambda }_1(\Omega )\le \cdots \le {\lambda }_k(\Omega )$ are the first $k$ eigenvalues on $\Omega$ of an operator in divergence form with Dirichlet boundary condition and with Hölder 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 $d^{\ast }\in \{5,6,7\}$.

33 pages