Symmetry breaking in the minimization of the first eigenvalue for the composite clamped punctured disk

Claudia Anedda; Fabrizio Cuccu

Applicationes Mathematicae (2015)

  • Volume: 42, Issue: 2-3, page 183-191
  • ISSN: 1233-7234

Abstract

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Let D₀=x∈ ℝ²: 0<|x|<1 be the unit punctured disk. We consider the first eigenvalue λ₁(ρ ) of the problem Δ² u =λ ρ u in D₀ with Dirichlet boundary condition, where ρ is an arbitrary function that takes only two given values 0 < α < β and is subject to the constraint ∫ D ₀ ρ d x = α γ + β ( | D ₀ | - γ ) for a fixed 0 < γ < |D₀|. We will be concerned with the minimization problem ρ ↦ λ₁(ρ). We show that, under suitable conditions on α, β and γ, the minimizer does not inherit the radial symmetry of the domain.

How to cite

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Claudia Anedda, and Fabrizio Cuccu. "Symmetry breaking in the minimization of the first eigenvalue for the composite clamped punctured disk." Applicationes Mathematicae 42.2-3 (2015): 183-191. <http://eudml.org/doc/279987>.

@article{ClaudiaAnedda2015,
abstract = {Let D₀=x∈ ℝ²: 0<|x|<1 be the unit punctured disk. We consider the first eigenvalue λ₁(ρ ) of the problem Δ² u =λ ρ u in D₀ with Dirichlet boundary condition, where ρ is an arbitrary function that takes only two given values 0 < α < β and is subject to the constraint $∫_\{D₀\} ρdx = αγ + β(|D₀|-γ)$ for a fixed 0 < γ < |D₀|. We will be concerned with the minimization problem ρ ↦ λ₁(ρ). We show that, under suitable conditions on α, β and γ, the minimizer does not inherit the radial symmetry of the domain.},
author = {Claudia Anedda, Fabrizio Cuccu},
journal = {Applicationes Mathematicae},
keywords = {eigenvalue problem; minimization; symmetry breaking; punctured disk},
language = {eng},
number = {2-3},
pages = {183-191},
title = {Symmetry breaking in the minimization of the first eigenvalue for the composite clamped punctured disk},
url = {http://eudml.org/doc/279987},
volume = {42},
year = {2015},
}

TY - JOUR
AU - Claudia Anedda
AU - Fabrizio Cuccu
TI - Symmetry breaking in the minimization of the first eigenvalue for the composite clamped punctured disk
JO - Applicationes Mathematicae
PY - 2015
VL - 42
IS - 2-3
SP - 183
EP - 191
AB - Let D₀=x∈ ℝ²: 0<|x|<1 be the unit punctured disk. We consider the first eigenvalue λ₁(ρ ) of the problem Δ² u =λ ρ u in D₀ with Dirichlet boundary condition, where ρ is an arbitrary function that takes only two given values 0 < α < β and is subject to the constraint $∫_{D₀} ρdx = αγ + β(|D₀|-γ)$ for a fixed 0 < γ < |D₀|. We will be concerned with the minimization problem ρ ↦ λ₁(ρ). We show that, under suitable conditions on α, β and γ, the minimizer does not inherit the radial symmetry of the domain.
LA - eng
KW - eigenvalue problem; minimization; symmetry breaking; punctured disk
UR - http://eudml.org/doc/279987
ER -

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