The Poles of the Resolvent for the Exterior Neumann Problem of Anisotropic Elasticity

Abstract

The poles of the resolvent and the asymptotic behavior of the local energy for the exterior Neumann problem of elastic wave equations are considered. For the most general class of anisotropic elastic media, the existence of the poles approaching the real axis is proved if the Rayleigh surface waves exist at least locally. The rate of their convergence to the real axis is estimated. Some results which show that the local energy hardly escapes from any neighborhood of the boundary are also presented. These results are considered as an influence of the existence of the Rayleigh surface waves.The local existence condition of the Rayleigh surface waves is given in terms of the surface impedance tensor, which is essentially equal to the principal part of the Neumann operator in the elliptic region. Unlike isotropic elastic media, the Rayleigh surface waves exist only locally for anisotropic elastic media. Nevertheless, the local existence of the Rayleigh surface waves is enough to prove the same results as thos...