Symmetry breaking in the minimization of the second eigenvalue for composite membranes
Published online by Cambridge University Press: 30 January 2015
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
- Information
- Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 1 , February 2015 , pp. 1 - 11
- Copyright
- Copyright © Royal Society of Edinburgh 2015
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