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.
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- bi-Laplace eigenvalues
- free plate