The limiting absorption principle for the two-dimensional inhomogeneous anisotropic elasticity system

Authors:
Gen Nakamura and Jenn-Nan Wang

Journal:
Trans. Amer. Math. Soc. **358** (2006), 147-165

MSC (2000):
Primary 35J55, 74G25, 74G30; Secondary 74B05, 74E10

Published electronically:
December 28, 2004

MathSciNet review:
2171227

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Abstract | References | Similar Articles | Additional Information

Abstract: In this work we establish the limiting absorption principle for the two-dimensional steady-state elasticity system in an inhomogeneous aniso- tropic medium. We then use the limiting absorption principle to prove the existence of a radiation solution to the exterior Dirichlet or Neumann boundary value problems for such a system. In order to define the radiation solution, we need to impose certain appropriate radiation conditions at infinity. It should be remarked that even though in this paper we assume that the medium is homogeneous outside of a large domain, it still preserves anisotropy. Thus the classical Kupradze's radiation conditions for the isotropic system are not suitable in our problem and new radiation conditions are required. The uniqueness of the radiation solution plays a key role in establishing the limiting absorption principle. To prove the uniqueness of the radiation solution, we make use of the unique continuation property, which was recently obtained by the authors. The study of this work is motivated by related inverse problems in the anisotropic elasticity system. The existence and uniqueness of the radiation solution are fundamental questions in the investigation of inverse problems.

**1.**Dang Ding Ang, Masaru Ikehata, Dang Duc Trong, and Masahiro Yamamoto,*Unique continuation for a stationary isotropic Lamé system with variable coefficients*, Comm. Partial Differential Equations**23**(1998), no. 1-2, 371–385. MR**1608540**, 10.1080/03605309808821349**2.**Krzysztof Chełmiński,*The principle of limiting absorption in elasticity*, Bull. Polish Acad. Sci. Math.**41**(1993), no. 1, 19–30. MR**1401881****3.**B. Dehman and L. Robbiano,*La propriété du prolongement unique pour un système elliptique. Le système de Lamé*, J. Math. Pures Appl. (9)**72**(1993), no. 5, 475–492 (French). MR**1239100****4.**G. F. D. Duff,*The Cauchy problem for elastic waves in an anistropic medium*, Philos. Trans. Roy. Soc. London Ser. A**252**(1960), 249–273. MR**0111293****5.***American Mathematical Society Translations. Series 2, Vol. 47: Thirteen papers on functional analysis and partial differential equations*, American Mathematical Society, Providence, R.I., 1965. MR**0188026****6.**I. M. Gel’fand and G. E. Shilov,*Generalized functions. Vol. I: Properties and operations*, Translated by Eugene Saletan, Academic Press, New York-London, 1964. MR**0166596****7.**Mariano Giaquinta,*Introduction to regularity theory for nonlinear elliptic systems*, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1993. MR**1239172****8.**Tosio Kato,*Perturbation theory for linear operators*, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR**1335452****9.**V. D. Kupradze, T. G. Gegelia, M. O. Basheleĭshvili, and T. V. Burchuladze,*Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity*, Translated from the second Russian edition, North-Holland Series in Applied Mathematics and Mechanics, vol. 25, North-Holland Publishing Co., Amsterdam-New York, 1979. Edited by V. D. Kupradze. MR**530377****10.**Rolf Leis,*Initial-boundary value problems in mathematical physics*, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986. MR**841971****11.**Walter Littman,*Decay at infinity of solutions to partial differential equations with constant coefficients*, Trans. Amer. Math. Soc.**123**(1966), 449–459. MR**0197951**, 10.1090/S0002-9947-1966-0197951-7**12.**Mutsuhide Matsumura,*Uniform estimates of elementary solutions of first order systems of partial differential equations*, Publ. Res. Inst. Math. Sci.**6**(1970), 293–305. MR**0284695****13.**G. Nakamura and J.-N. Wang,*Unique continuation for the two-dimensional anisotropic elasticity system and its applications to inverse problems*. Submitted.**14.**D. Natroshvili,*Two-dimensional steady-state oscillation problems of anisotropic elasticity*, Georgian Math. J.**3**(1996), no. 3, 239–262. MR**1388672**, 10.1007/BF02280007**15.**John R. Schulenberger and Calvin H. Wilcox,*A Rellich uniqueness theorem for steady-state wave propagation in inhomogeneous anisotropic media*, Arch. Rational Mech. Anal.**41**(1971), 18–45. MR**0274954****16.**John R. Schulenberger and Calvin H. Wilcox,*The limiting absorption principle and spectral theory for steady-state wave propagation in inhomogeneous anisotropic media*, Arch. Rational Mech. Anal.**41**(1971), 46–65. MR**0274955****17.**Norbert Weck,*Außenraumaufgaben in der Theorie stationärer Schwingungen inhomogener elastischer Körper*, Math. Z.**111**(1969), 387–398 (German). MR**0263295****18.**Norbert Weck,*Unique continuation for systems with Lamé principal part*, Math. Methods Appl. Sci.**24**(2001), no. 9, 595–605. MR**1834916**, 10.1002/mma.231**19.**Calvin H. Wilcox,*Wave operators and asymptotic solutions of wave propagation problems of classical physics*, Arch. Rational Mech. Anal.**22**(1966), 37–78. MR**0199531****20.**Calvin H. Wilcox,*Steady-state wave propagation in homogeneous anisotropic media*, Arch. Rational Mech. Anal.**25**(1967), 201–242. MR**0225528**

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Additional Information

**Gen Nakamura**

Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan

Email:
gnaka@math.sci.hokudai.ac.jp

**Jenn-Nan Wang**

Affiliation:
Department of Mathematics, National Taiwan University, Taipei 106, Taiwan

Email:
jnwang@math.ntu.edu.tw

DOI:
https://doi.org/10.1090/S0002-9947-04-03609-8

Keywords:
Limiting absorption principle,
anisotropic elasticity system,
radiation conditions

Received by editor(s):
September 15, 2003

Received by editor(s) in revised form:
January 5, 2004

Published electronically:
December 28, 2004

Additional Notes:
The first author was partially supported by Grant-in-Aid for Scientific Research (B)(2) (No.14340038) of the Japan Society for the Promotion of Science

The second author was partially supported by the National Science Council of Taiwan

Article copyright:
© Copyright 2004
American Mathematical Society