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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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: 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.


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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: http://dx.doi.org/10.1090/S0002-9947-04-03609-8
PII: S 0002-9947(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