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). DOI: 10.57262/ade/1465912586

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

Download Citation

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. https://doi.org/10.57262/ade/1465912586

Information

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

zbMATH: 1375.35543
MathSciNet: MR3513121
Digital Object Identifier: 10.57262/ade/1465912586

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

Rights: Copyright © 2016 Khayyam Publishing, Inc.

Vol.21 • No. 9/10 • September/October 2016
Back to Top