Carleman Estimate and Decay Rate of the Local Energy for the Neumann Problem of Elasticity
The asymptotic behavior of the local energy and the poles of the resolvent (scattering poles) associated to the elasticity operator in the exterior of an arbitrary obstacle with Neumann or Dirichlet boundary conditions are considered. We prove that there exists an exponentially small neighborhood of the real axis free of resonances. Consequently we prove that for regular data, the energy decays at least as fast as the inverse of the logarithm of the time. According to Stephanov—Vodev (, ), our results are optimal in the case of a Neumann boundary condition, even when the obstacle is a ball of ℝ3.
The fundamental difference between our case and the case of the scalar laplacian (see Burq ) is that the phenomenon of Rayleigh waves is connected to the failure of the Lopatinskii condition.
Key wordsCarleman estimate resonances energy decay elasticity system
AMS Subject Classification35 P20 35 L20 73 CO2
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