An isoperimetric inequality for fundamental tones of free plates with nonzero Poisson's ratio. Issue 8 (2nd August 2016)

• Record Type: Journal Article
• Authors: Chasman, L.M.
• Is Part Of: Applicable analysis. Volume 95:Issue 8(2016)
• Page Start: 1700
• Page End: 1735
• Publication Date: 2016-08-02
• Subjects: isoperimetric -- free plate -- bi-Laplace -- bi-Laplace eigenvalues; Primary: 35P15 -- Secondary: 35J40 -- 35J35; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2015.1068299
• Languages: English
• ISSNs: 0003-6811

Abstract

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.