Misc.

2006

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

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
  • Mishio Kawashita
  • ,
  • Wakako Kawashita
  • ,
  • Hideo Soga

Volume
358
Number
12
First page
5319
Last page
5350
Language
English
Publishing type
DOI
10.1090/S0002-9947-06-04244-9
Publisher
AMER MATHEMATICAL SOC

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. at 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.

Link information
DOI
https://doi.org/10.1090/S0002-9947-06-04244-9
Web of Science
https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcAuth=JSTA_CEL&SrcApp=J_Gate_JST&DestLinkType=FullRecord&KeyUT=WOS:000242401400006&DestApp=WOS_CPL
ID information
  • DOI : 10.1090/S0002-9947-06-04244-9
  • ISSN : 0002-9947
  • Web of Science ID : WOS:000242401400006

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