Symmetry breaking in the minimization of the second eigenvalue for composite membranes. Issue 1 (30th January 2015) - Anedda, Claudia; Cuccu, Fabrizio
Abstract : 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.