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L. M. Chasman

Publications and source records attributed to L. M. Chasman.

5 recordsLinked to original sources
arXiv2018

On Clamped Plates with Log-Convex Density

We consider the analogue of Rayleigh's conjecture for the clamped plate in Euclidean space weighted by a log-convex density. We show that the lowest eigenvalue of the bi-Laplace operator with drift in a given domain is bounded below by a constant $C(V,n)$ times the lowest eigenvalue of a centered ball of the same volume; the constant depends on the volume $V$ of the domain and the dimension $n$ of the ambient space. Our result is driven by a comparison theorem in the spirit of Talenti, and the constant $C(V,n)$ is defined in terms of a minimization problem following the work of Ashbaugh and Benguria. When the density is an "anti-Gaussian," we estimate $C(V,n)$ using a delicate analysis that involves confluent hypergeometric functions, and we illustrate numerically that $C(V,n)$ is close to $1$ for low dimensions.

math.SP↗
arXiv2010

An isoperimetric inequality for the fundamental tone of free plates

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 $|D^2u|^2$. 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.

math.SP↗
arXiv2010

Vibrational modes of circular free plates under tension

The vibrational frequencies of a plate under tension are given by the eigenvalues $ω$ of the equation $Δ^2u-τΔu=ωu$. This paper determines the eigenfunctions and eigenvalues of this bi-Laplace problem on the ball under natural (free) boundary conditions. In particular, the fundamental modes --- the eigenfunctions of the lowest nonzero eigenvalue --- are identified and found to have simple angular dependence.

math.AP↗
arXiv2010

An Isoperimetric Inequality for Fundamental Tones of Free Plates

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 $|D^2u|^2$. 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.

math.SP↗