Communications in Mathematical Physics

, Volume 303, Issue 2, pp 421–449 | Cite as

An Isoperimetric Inequality for Fundamental Tones of Free Plates

  • L. M. ChasmanEmail author


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 ΔΔuτΔu = 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.


Unit Ball Fundamental Mode Trial Function Isoperimetric Inequality Natural Boundary Condition 
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Authors and Affiliations

  1. 1.Knox CollegeGalesburgU.S.A.

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