An Isoperimetric Inequality for Fundamental Tones of Free Plates

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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 {pipe}D2u{pipe}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.

Original languageEnglish (US)
Pages (from-to)421-449
Number of pages29
JournalCommunications in Mathematical Physics
Issue number2
StatePublished - Apr 2011
Externally publishedYes


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