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Scattering theory for the elastic wave equation in perturbed half-spaces
Author(s):
Mishio
Kawashita;
Wakako
Kawashita;
Hideo
Soga
Journal:
Trans. Amer. Math. Soc.
358
(2006),
5319-5350.
MSC (2000):
Primary 35L20, 35P25, 74B05
Posted:
July 25, 2006
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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.
References:
-
- 1.
- J.D. Achenbach, ``Wave propagation in elastic solids'', North-Holland, New York, 1973.
- 2.
- Y. Dermenjian and J. Guillot, `` Scattering of elastic waves in a perturbed isotropic half space with a free boundary. The limiting absorption principle", Math. Meth. Appl. Sci. 10 (1988), 87-124. MR 0937415 (89i:35109)
- 3.
- H. Ito `` Extended Korn's inequalities and associated best possible constants", J. Elasticity 24 (1990), 43-78. MR 1086253 (91k:73025)
- 4.
- M. Ikawa `` Scattering Theory" (in Japanese) Iwanami Shoten, 1999.
- 5.
- M. Kawashita, `` On the decay rate of local energy for the elastic wave equation", Osaka J. Math. 30 (1993), 813-837. MR 1250785 (95b:35122)
- 6.
- M. Kawashita, `` Another proof of the representation formula of the scattering kernel for the elastic wave equation", Tsukuba J. Math. 18 (1994), 351-369. MR 1305819 (96a:35141)
- 7.
- M. Kawashita, W. Kawashita and H. Soga, `` Relation between scattering theories of the Wilcox and Lax-Phillips types and a concrete construction of the translation representation", Comm. P. D. E. 28 (2003), 1437-1470. MR 1998943 (2004f:47017)
- 8.
- M. Kawashita and W. Kawashita `` Analyticity of the resolvent for elastic waves in a perturbed isotropic half space", Math. Nachr. 278 (2005), 1163-1179. MR 2155967
- 9.
- P. D. Lax and R. S. Phillips, ``Scattering theory'', Academic Press, New York, 1967. MR 1037774 (90k:35005) (review of 2nd edition)
- 10.
- P. D. Lax and R. S. Phillips, `` Scattering theory for the acoustic equation in an even number of space dimensions", Indiana Univ. Math. J. 22 (1972), 101-134. MR 0304882 (46:4014)
- 11.
- A. Majda, `` A representation formula for the scattering operator and the inverse problem for arbitrary bodies", Comm. Pure Appl. Math. 30 (1977), 165-194. MR 0435625 (55:8583)
- 12.
- R. Melrose, `` Singularities and energy decay in acoustical scattering", Duke Math. J. 46 (1979), 43-59. MR 0523601 (80h:35104)
- 13.
- C. S. Morawetz, `` Exponential decay of solutions of the wave equation", Comm. Pure Appl. Math. 19 (1966), 439-444. MR 0204828 (34:4664)
- 14.
- R. S. Phillips, `` Scattering theory for the wave equation with a short range perturbation", Indiana Univ. Math. J. 31 (1982), 221-229. MR 0667785 (84i:35113)
- 15.
- R. Seeley, `` Extension of
 functions defined in a half space", Proc. Amer. Math. Soc. 15 (1964), 625-626. MR 0165392 (29:2676) - 16.
- Y. Shibata and H. Soga, `` Scattering theory for the elastic wave equation", Publ. RIMS Kyoto Univ. 25 (1989), 861-887. MR 1045996 (91f:35204)
- 17.
- H. Soga, `` Singularities of the scattering kernel for convex obstacles", J. Math. Kyoto Univ. 22 (1983), 729-765. MR 0685528 (84c:35085)
- 18.
- H. Soga, `` Representation of the scattering kernel for the elastic wave equation and singularities of the back-scattering", Osaka J. Math. 29 (1992), 809-836. MR 1192742 (94b:35199)
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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:
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
Posted:
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 of article:
Copyright
2006,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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