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.

**1.**J.D. Achenbach,*``Wave propagation in elastic solids''*, North-Holland, New York, 1973.**2.**Yves Dermenjian and Jean-Claude Guillot,*Scattering of elastic waves in a perturbed isotropic half space with a free boundary. The limiting absorption principle*, Math. Methods Appl. Sci.**10**(1988), no. 2, 87–124. MR**937415**, 10.1002/mma.1670100202**3.**Hiroya Ito,*Extended Korn’s inequalities and the associated best possible constants*, J. Elasticity**24**(1990), no. 1-3, 43–78. MR**1086253**, 10.1007/BF00115553**4.**M. Ikawa*`` Scattering Theory"*(in Japanese) Iwanami Shoten, 1999.**5.**Mishio Kawashita,*On the decay rate of local energy for the elastic wave equation*, Osaka J. Math.**30**(1993), no. 4, 813–837. MR**1250785****6.**Mishio Kawashita,*Another proof of the representation formula of the scattering kernel for the elastic wave equation*, Tsukuba J. Math.**18**(1994), no. 2, 351–369. MR**1305819****7.**Mishio Kawashita, Wakako Kawashita, and Hideo Soga,*Relation between scattering theories of the Wilcox and Lax-Phillips types and a concrete construction of the translation representation*, Comm. Partial Differential Equations**28**(2003), no. 7-8, 1437–1470. MR**1998943**, 10.1081/PDE-120024374**8.**Mishio Kawashita and Wakako Kawashita,*Analyticity of the resolvent for elastic waves in a perturbed isotropic half space*, Math. Nachr.**278**(2005), no. 10, 1163–1179. MR**2155967**, 10.1002/mana.200310300**9.**Peter D. Lax and Ralph S. Phillips,*Scattering theory*, 2nd ed., Pure and Applied Mathematics, vol. 26, Academic Press, Inc., Boston, MA, 1989. With appendices by Cathleen S. Morawetz and Georg Schmidt. MR**1037774****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**(1972/73), 101–134. MR**0304882****11.**Andrew Majda,*A representation formula for the scattering operator and the inverse problem for arbitrary bodies*, Comm. Pure Appl. Math.**30**(1977), no. 2, 165–194. MR**0435625****12.**R. B. Melrose,*Singularities and energy decay in acoustical scattering*, Duke Math. J.**46**(1979), no. 1, 43–59. MR**523601****13.**Cathleen S. Morawetz,*Exponential decay of solutions of the wave equation*, Comm. Pure Appl. Math.**19**(1966), 439–444. MR**0204828****14.**Ralph S. Phillips,*Scattering theory for the wave equation with a short range perturbation*, Indiana Univ. Math. J.**31**(1982), no. 5, 609–639. MR**667785**, 10.1512/iumj.1982.31.31045**15.**R. T. Seeley,*Extension of 𝐶^{∞} functions defined in a half space*, Proc. Amer. Math. Soc.**15**(1964), 625–626. MR**0165392**, 10.1090/S0002-9939-1964-0165392-8**16.**Yoshihiro Shibata and Hideo Soga,*Scattering theory for the elastic wave equation*, Publ. Res. Inst. Math. Sci.**25**(1989), no. 6, 861–887. MR**1045996**, 10.2977/prims/1195172509**17.**Hideo Soga,*Singularities of the scattering kernel for convex obstacles*, J. Math. Kyoto Univ.**22**(1982/83), no. 4, 729–765. MR**685528****18.**Hideo Soga,*Representation of the scattering kernel for the elastic wave equation and singularities of the back-scattering*, Osaka J. Math.**29**(1992), no. 4, 809–836. MR**1192742**

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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

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.