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.


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