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 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 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:
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