An Isoperimetric Inequality for Fundamental Tones of Free Plates

Details An Isoperimetric Inequality for Fundamental Tones of Free Plates An Isoperimetric Inequality for Fundamental Tones of Free Plates

2010-03-31

arxiv.org/abs/1004.0016v1

Mathematics

Spectral Theory

PhD thesis. Papers are in preparation

Scientific paper

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 $\tau>0$, the free plate eigenvalues $\omega$ and eigenfunctions $u$ are determined by the equation $\Delta\Delta u-\tau\Delta u = \omega u$ 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 $|D^2u|^2$. 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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