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

Let Ω ⊂ ℝ N be an open bounded connected set. We consider the eigenvalue problem –Δ u = λ ρu 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 ∫ Ω ρ d x = 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 k th 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.