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28 March 2024
 
  » arxiv » math.AP/9912116

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Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes
S. Chanillo ; D. Grieser ; M. Imai ; K. Kurata ; I. Ohnishi ;
Date 15 Dec 1999
Subject Analysis of PDEs; Optimization and Control MSC-class: 49Q10 (Primary), 35B99, 35R35, 35P30 (Secondary) | math.AP math.OC
AbstractWe consider the following eigenvalue optimization problem: Given a bounded domain $OmegasubsetR^n$ and numbers $alphageq 0$, $Ain [0,|Omega|]$, find a subset $DsubsetOmega$ of area $A$ for which the first Dirichlet eigenvalue of the operator $-Delta + alpha chi_D$ is as small as possible. We prove existence of solutions and investigate their qualitative properties. For example, we show that for some symmetric domains (thin annuli and dumbbells with narrow handle) optimal solutions must possess fewer symmetries than $Omega$; on the other hand, for convex $Omega$ reflection symmetries are preserved. Also, we present numerical results and formulate some conjectures suggested by them.
Source arXiv, math.AP/9912116
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