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September/October 2016 On the spectrum of an elastic solid with cusps
Vladimir Kozlov, Sergei A. Nazarov
Adv. Differential Equations 21(9/10): 887-944 (September/October 2016).

Abstract

The spectral problem of anisotropic elasticity with traction-free boundary conditions is considered in a bounded domain with a spatial cusp having its vertex at the origin and given by triples $(x_1,x_2,x_3)$ such that $x_3^{-2}(x_1,x_2) \in \omega$, where $\omega$ is a two-dimensional Lipschitz domain with a compact closure. We show that there exists a threshold $\lambda_\dagger>0$ expressed explicitly in terms of the elasticity constants and the area of $\omega$ such that the continuous spectrum coincides with the half-line $[\lambda_\dagger,\infty)$, whereas the interval $[0,\lambda_\dagger)$ contains only the discrete spectrum. The asymptotic formula for solutions to this spectral problem near cusp's vertex is also derived. A principle feature of this asymptotic formula is the dependence of the leading term on the spectral parameter.

Citation

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Vladimir Kozlov. Sergei A. Nazarov. "On the spectrum of an elastic solid with cusps." Adv. Differential Equations 21 (9/10) 887 - 944, September/October 2016.

Information

Published: September/October 2016
First available in Project Euclid: 14 June 2016

zbMATH: 1375.35543
MathSciNet: MR3513121

Subjects:
Primary: 35J44 , 35P05 , 35P15 , 35Q72 , 74G55

Rights: Copyright © 2016 Khayyam Publishing, Inc.

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Vol.21 • No. 9/10 • September/October 2016
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