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

Published online by Cambridge University Press:  30 January 2015

Claudia Anedda
Affiliation:
Mathematics and Computer Science Department, University of Cagliari, Via Università 40, 09124 Cagliari, Italy, ([email protected]; [email protected])
Fabrizio Cuccu
Affiliation:
Mathematics and Computer Science Department, University of Cagliari, Via Università 40, 09124 Cagliari, Italy, ([email protected]; [email protected])

Abstract

Let Ω ⊂ ℝN 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 ⟼ λk(ρ), where λk 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.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2015 

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