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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Scattering theory for the elastic wave equation in perturbed half-spaces


Authors: Mishio Kawashita, Wakako Kawashita and Hideo Soga
Journal: Trans. Amer. Math. Soc. 358 (2006), 5319-5350
MSC (2000): Primary 35L20, 35P25, 74B05
Published electronically: July 25, 2006
MathSciNet review: 2238918
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Abstract: In this paper we consider the linear elastic wave equation with the free boundary condition (the Neumann condition), and formulate a scattering theory of the Lax and Phillips type and a representation of the scattering kernel. We are interested in surface waves (the Rayleigh wave, etc.) connected closely with situations of boundaries, and make the formulations intending to extract this connection.

The half-space is selected as the free space, and making dents on the boundary is considered as a perturbation from the flat one. Since the lacuna property for the solutions in the outgoing and incoming spaces does not hold because of the existence of the surface waves, instead of it, certain decay estimates for the free space solutions and a weak version of the Morawetz arguments are used to formulate the scattering theory.

We construct the representation of the scattering kernel with outgoing scattered plane waves. In this step, again because of the existence of the surface waves, we need to introduce new outgoing and incoming conditions for the time dependent solutions to ensure uniqueness of the solutions. This introduction is essential to show the representation by reasoning similar to the case of the reduced wave equation.


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Additional Information

Mishio Kawashita
Affiliation: Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526 Japan
Email: kawasita@math.sci.hiroshima-u.ac.jp

Wakako Kawashita
Affiliation: Kagamiyama 2-360-2-1-303 Higashi-Hiroshima, 739-0046 Japan
Email: adt42760@rio.odn.ne.jp

Hideo Soga
Affiliation: Faculty of Education, Ibaraki University, Mito, Ibaraki, 310-8512, Japan
Email: soga@mx.ibaraki.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04244-9
PII: S 0002-9947(06)04244-9
Keywords: Scattering theory, distorted plane waves, scattering kernel, elastic wave equations, the Rayleigh wave
Received by editor(s): September 3, 2004
Published electronically: July 25, 2006
Additional Notes: The first author was partly supported by Grant-in-Aid for Science Research (C)(2) 16540156 from JSPS
The second author was partly supported by Grant-in-Aid for Science Research (C) 1554015 from JSPS
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.