Advances in Differential Equations

On the spectrum of an elastic solid with cusps

Vladimir Kozlov and Sergei A. Nazarov

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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

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

First available in Project Euclid: 14 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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