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

Abstract

In this paper we consider minimizers of the functionalmin⁡{λ1(Ω)+⋯+λk(Ω)+Λ|Ω|,:Ω⊂D open} where D⊂Rd is a bounded open set and where 0<λ1(Ω)≤⋯≤λk(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 C1,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}.