## Asymptotics of the number of Rayleigh resonances.(English)Zbl 0890.35098

Resonances associated to the Neumann problem in linear elasticity are studied. It is well-known that for this problem there are surface waves called Rayleigh waves moving with a speed $$c_R>0$$ strictly less than the two speeds in the exterior domain. For a class of obstacles including strictly convex ones the following asymptotic for the number, $$N(r)$$, of the resonances generated by the Rayleigh waves in the disk $$\{z\in\mathbb{C}:|z|\leq r\}$$ is obtained: $N(r)= \tau_n c^{-n+1}_R \text{Vol}(\Gamma) r^{n- 1}+ O(r^{n- 2}),\quad r\to\infty.$ Here, $$\tau_n= (2\pi)^{- n+1}\text{Vol}\{x\in \mathbb{R}^{n- 1}:|x|\leq 1\}$$, $$\Gamma$$ is the boundary of the obstacle, and $$n$$ is the space dimension.

### MSC:

 35P25 Scattering theory for PDEs 35J15 Second-order elliptic equations 47F05 General theory of partial differential operators

### Keywords:

asymptotic formula; Rayleigh waves; obstacles
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