An Isoperimetric Inequality for Fundamental Tones of Free Plates

TL;DR

This paper proves that among all plates with the same area, the circular shape maximizes the fundamental tone, extending isoperimetric inequalities to the eigenvalues of free plates governed by a biharmonic operator.

Contribution

The paper establishes a new isoperimetric inequality for the fundamental eigenvalue of free plates, identifying the ball as the maximizer, using adapted methods from membrane problems.

Findings

The ball maximizes the fundamental tone among plates of equal area.

Eigenvalues are characterized by a biharmonic equation with natural boundary conditions.

Explicit solutions involve Bessel functions for the fundamental modes.

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 $|D^2u|^2$. 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.