An isoperimetric inequality for the fundamental tone of free plates

TL;DR

This paper proves that among all free plates of a fixed area, the shape that maximizes the fundamental tone (first nonzero eigenvalue) is a ball, extending isoperimetric inequalities to plate vibrations.

Contribution

It establishes a new isoperimetric inequality for the fundamental eigenvalue of free plates, using adapted methods from membrane problems and Bessel function solutions.

Findings

The ball maximizes the fundamental tone among free plates of fixed area.

Eigenvalues are characterized by a biharmonic-type PDE with natural boundary conditions.

The proof adapts Weinberger's method and involves Bessel function solutions.

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.