An isoperimetric inequality for the fundamental tone of free plates

Details An isoperimetric inequality for the fundamental tone of free plates An isoperimetric inequality for the fundamental tone of free plates

2010-04-19

arxiv.org/abs/1004.3318v2

Mathematics

Spectral Theory

27 pages. 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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