An isoperimetric inequality for fundamental tones of free plates with nonzero Poisson’s ratio

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.

Keywords:

• isoperimetric
• free plate
• bi-Laplace
• bi-Laplace eigenvalues

AMS Subject Classifications:

• Primary: 35P15
• Secondary: 35J40
• 35J35

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.

Additional information

Funding

This research was partially supported by the University of Minnesota’s Faculty Development Single Semester Leave.