Distribution of Resonances and Decay Rate of the Local Energy for the Elastic Wave Equation |
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Authors: | Mourad Bellassoued |
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Institution: | (1) Université de Paris Sud, Mathématiques, Bat. 425, 91405 Orsay Cedex, France, FR |
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Abstract: | We study resonances (scattering poles) associated to the elasticity operator in the exterior of an arbitrary obstacle with
Neumann or Dirichlet boundary conditions. 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 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 ℝ3. 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 Lopatinskii condition.
Received: 22 February 2000 / Accepted: 28 June 2000 |
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