An Isoperimetric Inequality for Fundamental Tones of Free Plates

L. M. Chasman

Research output: Contribution to journalArticlepeer-review

23 Scopus citations

Abstract

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 {pipe}D2u{pipe}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.

Original languageEnglish (US)
Pages (from-to)421-449
Number of pages29
JournalCommunications in Mathematical Physics
Volume303
Issue number2
DOIs
StatePublished - Apr 2011
Externally publishedYes

Fingerprint

Dive into the research topics of 'An Isoperimetric Inequality for Fundamental Tones of Free Plates'. Together they form a unique fingerprint.

Cite this