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Communications in Mathematical Physics

, Volume 214, Issue 2, pp 315–337 | Cite as

Symmetry Breaking and Other Phenomena in the Optimization of Eigenvalues for Composite Membranes

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

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+αχ 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 Ω on the other hand, for convex Ω reflection symmetries are preserved.

Also, we present numerical results and formulate some conjectures suggested by them.

Keywords

Reflection Bounded Domain Composite Membrane Qualitative Property Symmetric Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • S. Chanillo
    • 1
  • D. Grieser
    • 2
  • M. Imai
    • 3
  • K. Kurata
    • 4
  • I. Ohnishi
    • 3
  1. 1.Department of Mathematics, Rutgers University, New Brunswick, NJ 98903, USA.¶E-mail: chanillo@math.rutgers.eduUS
  2. 2.Institut für Mathematik, Humboldt-Universität Berlin, Unter den Linden 6, 10099 Berlin, Germany.¶E-mail: grieser@mathematik.hu-berlin.deDE
  3. 3.Department of Information mathematics and Computer sciences, University of Electro-Communications, Chofu-ga-oka 1-5-1, Chofu-shi, Tokyo, Japan. E-mail: imai-m@kenks.im.uec.jp; ohnishi@im.uec.ac.jpJP
  4. 4.Department of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji-shi, Tokyo, Japan. E-mail: kurata@comp.metro-u.ac.jpJP

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