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Uniqueness of the density in an inverse problem for isotropic elastodynamics


Author: Lizabeth V. Rachele
Journal: Trans. Amer. Math. Soc. 355 (2003), 4781-4806
MSC (2000): Primary 35R30
DOI: https://doi.org/10.1090/S0002-9947-03-03268-9
Published electronically: July 28, 2003
MathSciNet review: 1997584
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Abstract: We consider the unique determination of the density of a nonhomogeneous, isotropic elastic object from measurements made at the surface. We model the behavior of the bounded, 3-dimensional object by the linear, hyperbolic system of operators for isotropic elastodynamics. The material properties of the object (its density and elastic properties) correspond to the smooth coefficients of these differential operators. The data for this inverse problem, in the form of the correspondence between applied surface tractions and resulting surface displacements, is modeled by the dynamic Dirichlet-to-Neumann map on a finite time interval. In an earlier paper we show that the speeds $c_{p/s}$ of (compressional and sheer) wave propagation through the object are uniquely determined by the Dirichlet-to-Neumann map. Here we extend that result by showing that the density is also determined in the interior by the Dirichlet-to-Neumann map in the case, for example, that $c_p = 2 c_s$ at only isolated points in the object. We use techniques from microlocal analysis and integral geometry to solve this fully three-dimensional problem.


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  • [De] N. Dencker. On the propagation of singularities for pseudodifferential operators of principal type, Ark. Mat. 20 (1), 1982, 23 - 60. MR 84k:58126
  • [Eg] Y. V. Egorov. Linear Differential Equations of Principal Type, Contemporary Soviet Math., New York, 1986. MR 88c:35003
  • [Gr-Sj] A. Grigis and J. Sjostrand. Microlocal Analysis for Differential Operators, London Math. Soc. Lecture Note Series 196, Cambridge University Press, 1994. MR 95d:35009
  • [Hö] L. Hörmander. The Analysis of Linear Partial Differential Operators, Vol. I, Springer-Verlag, Berlin, 1983. MR 85g:35002a
  • [I-N-Y] M. Ikehata, G. Nakamura, and M. Yamamoto. Uniqueness in inverse problems for the isotropic Lamé system, J. Math. Sci. Univ. Tokyo 5 (4), 1998, 627 - 692. MR 2000i:35213
  • [Pr] M. H. Protter. Unique continuation for elliptic equations, Trans. Amer. Math. Soc. 95 (1), 1960, 81 - 91. MR 22:3871
  • [R I] L. Rachele. An Inverse Problem in Elastodynamics: Uniqueness of the wave speeds in the interior, J. Diff. Eqs. 162 (2), 2000, 300 - 325. MR 2001e:35177
  • [R II] L. Rachele. Boundary determination for an inverse problem in elastodynamics, Comm. Partial Diff. Eq. 25 (11-12), 2000, 1951 - 1996. MR 2001m:35322
  • [R III] L. Rachele. Uniqueness in inverse problems for elastic media with residual stress. To appear in Comm. Partial Diff. Eq.
  • [Ra-Sy] Rakesh and W. W. Symes. Uniqueness for an inverse problem for the wave equation, Comm. Partial Diff. Eq. 13 (1), 1988, 87 - 96. MR 89f:35208
  • [Sh I] V. Sharafutdinov. Integral Geometry of Tensor Fields, Inverse and Ill-Posed Problems Series. VSP, Utrecht, 1994. MR 97h:53077
  • [Sh II] V. Sharafutdinov. Integral geometry of a tensor field on a manifold with upper-bounded curvature, Siberian Math. J. 33 (3), 1992, 524-533. MR 94d:53116
  • [S-U] J. Sylvester and G. Uhlmann. Inverse problems in anisotropic media. Contemporary Mathematics 122, 1991, 105 - 117. MR 92k:35289
  • [Tr] F. Treves. Introduction to pseudodifferential and Fourier integral operators, vol. II, Plenum Press, New York, 1980. MR 82i:58068
  • [U] G. Uhlmann. Inverse boundary value problems and applications, Méthodes semi-classiques, Vol. 1 (Nantes, 1991), Asterisque 207 (6), 1992, 153 - 211. MR 94e:35146

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Additional Information

Lizabeth V. Rachele
Affiliation: Department of Mathematics, Tufts University, Medford, Massachusetts 02155
Address at time of publication: Department of Mathematics and Statistics, University at Albany, SUNY, Albany, New York 12222
Email: lrachele@math.albany.edu

DOI: https://doi.org/10.1090/S0002-9947-03-03268-9
Received by editor(s): June 11, 2001
Published electronically: July 28, 2003
Additional Notes: The author was partially supported by U.S. National Science Foundation grant 9801664 (9996350)
Article copyright: © Copyright 2003 American Mathematical Society

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