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Carleman Estimate and Decay Rate of the Local Energy for the Neumann Problem of Elasticity

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Carleman Estimates and Applications to Uniqueness and Control Theory

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 46))

Abstract

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 ([17], [18]), 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 [1]) is that the phenomenon of Rayleigh waves is connected to the failure of the Lopatinskii condition.

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Authors and Affiliations

  1. Mathématiques, Bât. 425, Université de Paris Sud, 91405, Orsay Cedex, France

    Mourad Bellassoued

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  1. Mourad Bellassoued
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Editors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, 56127, Pisa, Italy

    Ferruccio Colombini

  2. Laboratoire de Mathématique, Université de Paris Sud—Orsay, 91405, Orsay, France

    Claude Zuily

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Bellassoued, M. (2001). Carleman Estimate and Decay Rate of the Local Energy for the Neumann Problem of Elasticity. In: Colombini, F., Zuily, C. (eds) Carleman Estimates and Applications to Uniqueness and Control Theory. Progress in Nonlinear Differential Equations and Their Applications, vol 46. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0203-5_2

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  • DOI: https://doi.org/10.1007/978-1-4612-0203-5_2

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6660-0

  • Online ISBN: 978-1-4612-0203-5

  • eBook Packages: Springer Book Archive

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