An Isoperimetric Inequality for Fundamental Tones of Free Plates
Laura Mercredi Chasman · arXiv (Cornell University) · 2010
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 $τ>0$, the free plate eigenvalues $ω$ and eigenfunctions $u$ are determined by the equation $ΔΔu-τΔu = ωu$ 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.