An isoperimetric inequality for the fundamental tone of free plates

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 $\tau >0$, 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 $|D^{2}u{|}^2$. 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.

27 pages. In preparation.

References in corpus (2)

• Vibrational modes of circular free plates under tension
• An Isoperimetric Inequality for Fundamental Tones of Free Plates

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