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An Isoperimetric Inequality for Fundamental Tones of Free Plates

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Abstract

We establish an isoperimetric inequality for the fundamental tone (first nonzero eigenvalue) of the free plate of a given area, proving the ball is maximal. Given τ > 0, the free plate eigenvalues ω and eigenfunctions u are determined by the equation ΔΔu − τΔu = ωu together with certain natural boundary conditions. The boundary conditions are complicated but arise naturally from the plate Rayleigh quotient, which contains a Hessian squared term |D 2 u|2.

We adapt Weinberger’s method from the corresponding free membrane problem, taking the fundamental modes of the unit ball as trial functions. These solutions are a linear combination of Bessel and modified Bessel functions.

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  1. Knox College, Galesburg, IL, 61401, U.S.A.

    L. M. Chasman

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Communicated by B. Simon

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Chasman, L.M. An Isoperimetric Inequality for Fundamental Tones of Free Plates. Commun. Math. Phys. 303, 421–449 (2011). https://doi.org/10.1007/s00220-010-1171-z

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