An Isoperimetric Inequality for Fundamental Tones of Free Plates
By L. Chasman, E. Kirr. 2010
57 citations recorded by the source corpus.
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 |D2u|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.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/1e798ba48da2a191c5bf5b60dc6fc2bb6bce5dfc
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In Applied Mathematics
In Computational Theory and Mathematics
In Mathematical Physics
Citations in the corpus
- A few shape optimization results for a biharmonic Steklov problem (2015, methodology): “Regarding Neumann boundary conditions, Chasman [14] proved that the ball is a maximizer for the fundamental tone.”
- Isoperimetric Inequalities for Eigenvalues of the Laplacian (2007, background): “In [28, 29], Chasman proves the following isoperimetric inequality, ω1(Ω) ≤ ω1(Ω), where Ω is a ball of the same volume as Ω (here, equality is attained if and only if Ω is a ball).”