An isoperimetric inequality for fundamental tones of free plates with nonzero Poisson’s ratio

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Abstract

We establish a partial generalization of a prior isoperimetric inequality for the fundamental tone (first nonzero eigenvalue) of the free plate to that of plates of nonzero Poisson’s ratio. Given a tension τ > 0 and a Poisson’s ratio σ , the free plate eigenvalues ω and eigenfunctions u are determined by the equation ΔΔu -τΔu = ωu together with certain natural boundary conditions which involve both τ and σ. The boundary conditions are complicated but arise naturally from the plate Rayleigh quotient. We prove the free plate isoperimetric inequality, previously shown in the σ = 0 case, holds for certain nonzero σ and positive τ. We conjecture that the inequality holds for all dimensions, τ > 0 , and relevant values of σ , and discuss numerical and analytic support of this conjecture.

Original languageEnglish (US)
Pages (from-to)1700-1735
Number of pages36
JournalApplicable Analysis
Volume95
Issue number8
DOIs
StatePublished - Aug 2 2016

Bibliographical note

Publisher Copyright:
© 2015 Informa UK Limited, trading as Taylor & Francis Group.

Keywords

  • bi-Laplace
  • bi-Laplace eigenvalues
  • free plate
  • isoperimetric

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