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

### Baptiste Trey

## Abstract

In this paper we consider minimizers of the functional

where $D⊂R_{d}$ is a bounded open set and where $0<λ_{1}(Ω)≤⋯≤λ_{k}(Ω)$ 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 $Ω_{⁎}$ have finite perimeter and that their free boundary $∂Ω_{⁎}∩D$ is composed of a *regular part*, which is locally the graph of a $C_{1,α}$-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 $d_{⁎}∈{5,6,7}$.

## Cite this article

Baptiste Trey, Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 5, pp. 1337–1371

DOI 10.1016/J.ANIHPC.2020.11.002