TY - JOUR

T1 - An Isoperimetric Inequality for Fundamental Tones of Free Plates

AU - Chasman, L. M.

PY - 2011/4

Y1 - 2011/4

N2 - 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.

AB - 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.

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U2 - 10.1007/s00220-010-1171-z

DO - 10.1007/s00220-010-1171-z

M3 - Article

AN - SCOPUS:79953044778

SN - 0010-3616

VL - 303

SP - 421

EP - 449

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -