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

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- by Mishio Kawashita, Wakako Kawashita and Hideo Soga PDF
- Trans. Amer. Math. Soc.
**358**(2006), 5319-5350 Request permission

## 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
- 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 - © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**358**(2006), 5319-5350 - MSC (2000): Primary 35L20, 35P25, 74B05
- DOI: https://doi.org/10.1090/S0002-9947-06-04244-9
- MathSciNet review: 2238918