Advances in Differential Equations

On the spectrum of an elastic solid with cusps

Vladimir Kozlov and Sergei A. Nazarov

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
Adv. Differential Equations Volume 21, Number 9/10 (2016), 887-944.

Dates
First available in Project Euclid: 14 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ade/1465912586

Mathematical Reviews number (MathSciNet)
MR3513121

Subjects
Primary: 35J44 35P05: General topics in linear spectral theory 35P15: Estimation of eigenvalues, upper and lower bounds 35Q72 74G55: Qualitative behavior of solutions

Citation

Kozlov, Vladimir; Nazarov, Sergei A. On the spectrum of an elastic solid with cusps. Adv. Differential Equations 21 (2016), no. 9/10, 887--944. https://projecteuclid.org/euclid.ade/1465912586.


Export citation