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Neumann resonances in linear elasticity for an arbitrary body

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Abstract

We study resonances (scattering poles) associated to the elasticity operator in the exterior of an arbitrary obstacle inR 3 with Neumann boundary conditions. We prove that there exists a sequence of resonances tending rapidly to the real axis.

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References

  • [CP] Cardoso, F., Popov, G.: Rayleigh quasimodes in linear elasticity. Comm. P.D.E.17, 1327–1367 (1992)

    Google Scholar 

  • [G] Gérard, C.: Asymptotique des poles de la matrice de scattering pour deux obstacles strictement convex. Bull. Soc. Math. France, Mémoire n. 31,116, 1988

  • [GK] Gohberg, I., Krein, M.: Introduction to the theory of linear non-selfadjoint operators. Providence, RI: AMS, 1969

    Google Scholar 

  • [I1] Ikawa, M.: On the poles of the scattering matrix for two strictly convex obstacles. J. Math. Kyoto Univ.23–1, 127–194 (1983)

    Google Scholar 

  • [I2] Ikawa, M.: Precise information on the poles of the scattering matrix for two strictly convex obstacles. J. Math. Kyoto Univ.27–1, 69–102 (1987)

    Google Scholar 

  • [I3] Ikawa, M.: Trapping obstacles with a sequence of poles of the scattering matrix converging to the real axis. Osaka J. Math.22, 657–689 (1985)

    Google Scholar 

  • [K] Kawashita, M.: On the local-energy decay property for the elastic wave equation with the Neumann boundary conditions. Duke Math. J.67, 333–351 (1992)

    Article  Google Scholar 

  • [L] Lazutkin, V.: Asymptotics of the eigenvalues of the Laplacian and quasimodes. Math. USSR Izvestija7, 439–466 (1973)

    Google Scholar 

  • [P] Popov, G.: Quasimodes for the Laplace operator and glancing hypersurfaces. In: M. Beals, R. Melrose, J. Rauch (eds.): Proceeding of Conference on Microlocal Analysis and nonlinear waves, Minnesota 1989, Berlin-Heidelberg-New York: Springer, 1991

    Google Scholar 

  • [SV1] Stefanov, P., Vodev, G.: Distribution of resonances for the Neumann problem in linear elasticity outside a ball. Ann. Inst. H. Poincaré (Physique Théorique)60, 303–321 (1994)

    Google Scholar 

  • [SV2] Stefanov, P., Vodev, G.: Distribution of resonances for the Neumann problem in linear elasticity outside a strictly convex body. Duke Math. J.78, 677–714 (1995)

    Article  Google Scholar 

  • [T] Taylor, M.: Rayleigh waves in linear elasticity as a propagation of singularities phenomenon. In: Proc. Conf. on P.D.E. and Geometry, New York: Marcel Dekker, 1979 pp. 273–291

    Google Scholar 

  • [Ti] Titchmarsh, E.C.: The Theory of Functions. Oxford: Oxford Univ. Press, 1968

    Google Scholar 

  • [Y] Yamamoto, K.: Singularities of solutions to the boundary value problems for elastic and Maxwell's equations. Japan J. Math.14, 119–163 (1988)

    Google Scholar 

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Communicated by B. Simon

Partly supported by BSF under grant MM 401.

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Stefanov, P., Vodev, G. Neumann resonances in linear elasticity for an arbitrary body. Commun.Math. Phys. 176, 645–659 (1996). https://doi.org/10.1007/BF02099253

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