Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes

S. Chanillo, D. Grieser, M. Imai, K. Kurata, I. Ohnishi

Research output: Contribution to journalArticle

64 Citations (Scopus)

Abstract

We consider the following eigenvalue optimization problem: Given a bounded domain Ω ⊂ ℝ and numbers α > 0, A ∈ [0, |Ω|], find a subset D ⊂ Ω of area A for which the first Dirichlet eigenvalue of the operator -Δ + αχD 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 Ω; on the other hand, for convex Ω reflection symmetries are preserved. Also, we present numerical results and formulate some conjectures suggested by them.

Original languageEnglish (US)
Pages (from-to)315-337
Number of pages23
JournalCommunications In Mathematical Physics
Volume214
Issue number2
DOIs
StatePublished - Nov 2000

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Eigenvalue Optimization
Thin Domains
Dirichlet Eigenvalues
Reflectional symmetry
First Eigenvalue
Qualitative Properties
Ring or annulus
Symmetry Breaking
Eigenvalue Problem
Existence of Solutions
Bounded Domain
broken symmetry
eigenvalues
Membrane
Optimal Solution
Composite
membranes
Optimization Problem
Eigenvalue
Symmetry

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Chanillo, S. ; Grieser, D. ; Imai, M. ; Kurata, K. ; Ohnishi, I. / Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes. In: Communications In Mathematical Physics. 2000 ; Vol. 214, No. 2. pp. 315-337.
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Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes. / Chanillo, S.; Grieser, D.; Imai, M.; Kurata, K.; Ohnishi, I.

In: Communications In Mathematical Physics, Vol. 214, No. 2, 11.2000, p. 315-337.

Research output: Contribution to journalArticle

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