### 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 + α_{χD} 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 language | English (US) |
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Pages (from-to) | 315-337 |

Number of pages | 23 |

Journal | Communications In Mathematical Physics |

Volume | 214 |

Issue number | 2 |

DOIs | |

State | Published - Nov 2000 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications In Mathematical Physics*,

*214*(2), 315-337. https://doi.org/10.1007/PL00005534

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*Communications In Mathematical Physics*, vol. 214, no. 2, pp. 315-337. https://doi.org/10.1007/PL00005534

**Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes.** / Chanillo, S.; Grieser, D.; Imai, M.; Kurata, K.; Ohnishi, I.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes

AU - Chanillo, S.

AU - Grieser, D.

AU - Imai, M.

AU - Kurata, K.

AU - Ohnishi, I.

PY - 2000/11

Y1 - 2000/11

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0034348254&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034348254&partnerID=8YFLogxK

U2 - 10.1007/PL00005534

DO - 10.1007/PL00005534

M3 - Article

AN - SCOPUS:0034348254

VL - 214

SP - 315

EP - 337

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -