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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
Article

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 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.

Keywords

Unit Ball Fundamental Mode Trial Function Isoperimetric Inequality Natural Boundary Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Knox CollegeGalesburgU.S.A.

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