An isoperimetric inequality for the fundamental tone of free plates

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 $\mathrm{\Delta }\mathrm{\Delta }u-\tau \mathrm{\Delta }u=\omega u$ 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 $\mathrm{\mid }{D}^{2}u{\mathrm{\mid }}^2$. 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.