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

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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.

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Authors and Affiliations

  1. Department of Mathematics, Rutgers University, New Brunswick, NJ 98903, USA.¶E-mail: chanillo@math.rutgers.edu, , , , , , US

    S. Chanillo

  2. Institut für Mathematik, Humboldt-Universität Berlin, Unter den Linden 6, 10099 Berlin, Germany.¶E-mail: grieser@mathematik.hu-berlin.de, , , , , , DE

    D. Grieser

  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.jp, , , , , , JP

    M. Imai & I. Ohnishi

  4. Department of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji-shi, Tokyo, Japan. E-mail: kurata@comp.metro-u.ac.jp, , , , , , JP

    K. Kurata

Authors
  1. S. Chanillo
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  2. D. Grieser
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  3. M. Imai
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  4. K. Kurata
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  5. I. Ohnishi
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Additional information

Received: 22 November 1999/ Accepted: 31 March 2000

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Chanillo, S., Grieser, D., Imai, M. et al. Symmetry Breaking and Other Phenomena in the Optimization of Eigenvalues for Composite Membranes. Commun. Math. Phys. 214, 315–337 (2000). https://doi.org/10.1007/PL00005534

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  • DOI: https://doi.org/10.1007/PL00005534

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