Symmetry breaking in the minimization of the second eigenvalue for composite membranes

Abstract

Let Ω ⊂ ℝ^N be an open bounded connected set. We consider the eigenvalue problem –Δu = λρu in Ω with Dirichlet boundary condition, where ρ is an arbitrary function that assumes only two given values 0 < α < β and is subject to the constraint ∫_Ωρ dx = αγ + β(|Ω| – γ) for a fixed 0 < γ < |Ω|. Cox and McLaughlin studied the optimization of the map ρ ⟼ λ_k(ρ), where λ_k is the kth eigenvalue. In this paper we focus our attention on the case when N ≥ 2, k = 2 and Ω is a ball. We show that, under suitable conditions on α, β and γ, the minimizers do not inherit radial symmetry.