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

 

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
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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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: http://dx.doi.org/10.1090/S0002-9947-03-03268-9
PII: S 0002-9947(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