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Unique continuation for the two-dimensional anisotropic elasticity system and its applications to inverse problems


Authors: Gen Nakamura and Jenn-Nan Wang
Journal: Trans. Amer. Math. Soc. 358 (2006), 2837-2853
MSC (2000): Primary 35B60, 74B05; Secondary 74G75
DOI: https://doi.org/10.1090/S0002-9947-06-03938-9
Published electronically: February 6, 2006
MathSciNet review: 2216248
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Abstract: Under some generic assumptions we prove the unique continuation property for the two-dimensional inhomogeneous anisotropic elasticity system. Having established the unique continuation property, we then investigate the inverse problem of reconstructing the inclusion or cavity embedded in a plane elastic body with inhomogeneous anisotropic medium by infinitely many localized boundary measurements.


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  • 1. D.D. Ang, M. Ikehata, D.D. Trong and M. Yamamoto, Unique continuation for a stationary isotropic Lamé system with varaiable coefficients, Comm. in PDE, 23 (1998), 371-385. MR 1608540 (98j:35049)
  • 2. G. Alessandrini and A. Morassi, Strong unique continuation for the Lamé system of elasticity, Comm. in PDE., 26 (2001), 1787-1810. MR 1865945 (2003j:35030)
  • 3. T. Carleman, Sur un problèm d'unicité pour les systemes d'équations aux dérivées partielles à deux varaibles indépendentes, Ark. Mat., Astr. Fys., 26B (1939), 1-9.
  • 4. B. Dehman and L. Rabbiano, La propriété du prolongement unique pour un système elliptique: le système de Lamé, J. Math. Pures Appl., 72 (1993), 475-492. MR 1239100 (94h:35051)
  • 5. A. Douglis, Uniqueness in Cauchy problem for elliptic systems of equations, Comm. Pure Appl. Math., 6 (1953), 291-298. MR 0064278 (16:257d)
  • 6. M. Gurtin, The Linear Theory of Elasticity, Mechanics of Solids, Vol. II (Thruesdell C., ed.), Springer-Verlag, Berlin, 1984.
  • 7. T. Hildebrandt and L. Graves, Implicit functions and their differentials in general analysis, Trans. Amer. Math. Soc., 29 (1927), 127-153. MR 1501380
  • 8. G. Hile and M. Protter, Unique continuation and the Cauchy problem for first order systems of partial differential equations, Comm. in PDE, 1 (1976), 437-465. MR 0481439 (58:1555)
  • 9. M. Ikehata, Reconstruction of the shape of the inclusion by boundary measurements, Comm. in PDE, 23 (1998), 1459-1474. MR 1642619 (99f:35222)
  • 10. M. Ikehata, The probe method and its applications, Inverse problems and related topics (Kobe, 1998), 57-68, Chapman and Hall/CRC Res. Notes Math., 419, Chapman & Hall/CRC, Boca Raton, FL, 2000. MR 1761338 (2001c:35245)
  • 11. M. Ikehata, G. Nakamura, and K. Tanuma, Identification of the shape of the inclusion in the anisotropic elastic body, Appl. Anal., 72 (1999), 17-26. MR 1775432 (2001d:74026)
  • 12. M. Ikehata and G. Nakamura, Reconstruction of cavity from boundary measurements, preprint.
  • 13. V. Isakov, On uniqueness of recovery of a discontinuous conductivity coefficients, Comm. Pure Appl. Math, 41 (1988), 865-877. MR 0951742 (90f:35205)
  • 14. C.-L. Lin, Strong unique continuation for an elasticity system with residual stress, Indiana Univ. Math. J., 53 (2004), 557-582. MR 2060045 (2005f:35033)
  • 15. C.-L. Lin and J.-N. Wang, Strong unique continuation for the Lamé system with Lipschitz coefficients, Math. Annalen, to appear.
  • 16. G. Nakamura, G. Uhlmann, and J.-N. Wang, Reconstruction of cracks in an anisotropic elastic medium, J. Math. Pures Appl., 82 (2003), 1251-1276. MR 2020922 (2005b:35299)
  • 17. G. Nakamura, G. Uhlmann, and J.-N. Wang, Reconstruction of cracks in an inhomogeneous anisotropic medium using point sources, Advances in Appl. Math., to appear.
  • 18. G. Nakamura and J.-N. Wang, Unique continuation for an elasticity system with residual stress and its applications, SIAM J. Math. Anal., 35 (2003), 304-317. MR 2001103 (2004k:35037)
  • 19. G. Nakamura and J.-N. Wang, The limiting absorption principle for the two-dimensional inhomogeneous anisotropic elasticity system, Trans. Amer. Math. Soc., 358 (2006), 147-165.
  • 20. G. Uhlmann, Developments in inverse problems since Calderon's foundational paper, Harmonic analysis and partial differential equations (Chicago, IL, 1996), 295-345, Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 1999. MR 1743870 (2000m:35181)
  • 21. N. Weck, Unique continuation for systems with Lamé principal part, Math. Methods Appl. Sci., 24 (2001), 595-605. MR 1834916 (2002f:35048)
  • 22. E. Zeidler, Nonlinear Function Analysis and its Applications, I. Fixed-Point Theorems, Springer-Verlag, New York, 1986. MR 0816732 (87f:47083)

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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/S0002-9947-06-03938-9
Keywords: Unique continuation, anisotropic elasticity system, inverse problems
Received by editor(s): January 5, 2004
Published electronically: February 6, 2006
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 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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