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

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.