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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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.
 1.
J.D. Achenbach, ``Wave propagation in elastic solids'', NorthHolland, New York, 1973.
 2.
Yves
Dermenjian and JeanClaude
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
(89i:35109), http://dx.doi.org/10.1002/mma.1670100202
 3.
Hiroya
Ito, Extended Korn’s inequalities and the associated best
possible constants, J. Elasticity 24 (1990),
no. 13, 43–78. MR 1086253
(91k:73025), http://dx.doi.org/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
(95b:35122)
 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
(96a:35141)
 7.
Mishio
Kawashita, Wakako
Kawashita, and Hideo
Soga, Relation between scattering theories of the Wilcox and
LaxPhillips types and a concrete construction of the translation
representation, Comm. Partial Differential Equations
28 (2003), no. 78, 1437–1470. MR 1998943
(2004f:47017), http://dx.doi.org/10.1081/PDE120024374
 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
(2006e:35322), http://dx.doi.org/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
(90k:35005)
 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
(46 #4014)
 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
(55 #8583)
 12.
R.
B. Melrose, Singularities and energy decay in acoustical
scattering, Duke Math. J. 46 (1979), no. 1,
43–59. MR
523601 (80h:35104)
 13.
Cathleen
S. Morawetz, Exponential decay of solutions of the wave
equation, Comm. Pure Appl. Math. 19 (1966),
439–444. MR 0204828
(34 #4664)
 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
(84i:35113), http://dx.doi.org/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
(29 #2676), http://dx.doi.org/10.1090/S00029939196401653928
 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
(91f:35204), http://dx.doi.org/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
(84c:35085)
 18.
Hideo
Soga, Representation of the scattering kernel for the elastic wave
equation and singularities of the backscattering, Osaka J. Math.
29 (1992), no. 4, 809–836. MR 1192742
(94b:35199)
 1.
 J.D. Achenbach, ``Wave propagation in elastic solids'', NorthHolland, 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), 87124. MR 0937415 (89i:35109)
 3.
 H. Ito `` Extended Korn's inequalities and associated best possible constants", J. Elasticity 24 (1990), 4378. 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), 813837. 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), 351369. MR 1305819 (96a:35141)
 7.
 M. Kawashita, W. Kawashita and H. Soga, `` Relation between scattering theories of the Wilcox and LaxPhillips types and a concrete construction of the translation representation", Comm. P. D. E. 28 (2003), 14371470. 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), 11631179. 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), 101134. 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), 165194. MR 0435625 (55:8583)
 12.
 R. Melrose, `` Singularities and energy decay in acoustical scattering", Duke Math. J. 46 (1979), 4359. MR 0523601 (80h:35104)
 13.
 C. S. Morawetz, `` Exponential decay of solutions of the wave equation", Comm. Pure Appl. Math. 19 (1966), 439444. 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), 221229. MR 0667785 (84i:35113)
 15.
 R. Seeley, `` Extension of functions defined in a half space", Proc. Amer. Math. Soc. 15 (1964), 625626. MR 0165392 (29:2676)
 16.
 Y. Shibata and H. Soga, `` Scattering theory for the elastic wave equation", Publ. RIMS Kyoto Univ. 25 (1989), 861887. MR 1045996 (91f:35204)
 17.
 H. Soga, `` Singularities of the scattering kernel for convex obstacles", J. Math. Kyoto Univ. 22 (1983), 729765. MR 0685528 (84c:35085)
 18.
 H. Soga, `` Representation of the scattering kernel for the elastic wave equation and singularities of the backscattering", Osaka J. Math. 29 (1992), 809836. MR 1192742 (94b:35199)
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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.
