  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.
