Abstract
We establish a partial generalization of a prior isoperimetric inequality for the fundamental tone (first nonzero eigenvalue) of the free plate to that of plates of nonzero Poisson’s ratio. Given a tension and a Poisson’s ratio
, the free plate eigenvalues
and eigenfunctions
are determined by the equation
together with certain natural boundary conditions which involve both
and
. The boundary conditions are complicated but arise naturally from the plate Rayleigh quotient. We prove the free plate isoperimetric inequality, previously shown in the
case, holds for certain nonzero
and positive
. We conjecture that the inequality holds for all dimensions,
, and relevant values of
, and discuss numerical and analytic support of this conjecture.
AMS Subject Classifications:
Acknowledgement
The author would like to thank Richard Laugesen for tirelessly offering advice on matters both mathematical and professional.
Notes
No potential conflict of interest was reported by the author.