## Advances in Differential Equations

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

### On the spectrum of an elastic solid with cusps

Vladimir Kozlov and Sergei A. Nazarov

#### 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

**Zentralblatt MATH identifier**

1375.35543

**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