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

L. M. Chasman

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    10 Scopus citations

    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

    Keywords

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

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