Scattering theory for the elastic wave equation in perturbed halfspaces
Authors:
Mishio Kawashita, Wakako Kawashita and Hideo Soga
Journal:
Trans. Amer. Math. Soc. 358 (2006), 53195350
MSC (2000):
Primary 35L20, 35P25, 74B05
Published electronically:
July 25, 2006
MathSciNet review:
2238918
Fulltext PDF Free Access
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Additional Information
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 halfspace 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, HigashiHiroshima, 7398526 Japan
Email:
kawasita@math.sci.hiroshimau.ac.jp
Wakako Kawashita
Affiliation:
Kagamiyama 236021303 HigashiHiroshima, 7390046 Japan
Email:
adt42760@rio.odn.ne.jp
Hideo Soga
Affiliation:
Faculty of Education, Ibaraki University, Mito, Ibaraki, 3108512, Japan
Email:
soga@mx.ibaraki.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002994706042449
PII:
S 00029947(06)042449
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 GrantinAid for Science Research (C)(2) 16540156 from JSPS
The second author was partly supported by GrantinAid 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.
