Distribution of resonances and decay rate of the local energy for the elastic wave equation

We study resonances (scattering poles) associated to the elasticity operato r in the exterior of an arbitrary obstacle with Neumann or Dirichlet bounda ry conditions. We prove that there exists an exponentially small neighborho od of the real axis free of resonances. Consequently we prove that for regu lar data, the energy for the elastic wave equation decays at least as fast as the inverse of the logarithm of time. According to Stefanov-Vodev ([SV1, SV2]), our results are optimal in the case of a Neumann boundary condition , even when the obstacle is a ball of R-3. The main difference between our case and the case of the scalar Laplacian (see Burq [Bu]) is the phenomenon of Rayleigh surface waves, which are connected to the failure of the Lopat inskii condition.