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

where $D\subset {\mathbb{R}}^d$ is a bounded open set and where $0<{\lambda }_1(\mathrm{\Omega })\le \cdots \le {\lambda }_k(\mathrm{\Omega })$ 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 ${\mathrm{\Omega }}^{\ast }$ have finite perimeter and that their free boundary $\partial {\mathrm{\Omega }}^{\ast }\cap D$ 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 $d^{\ast }\in \{5,6,7\}$.