Uniqueness of the density in an inverse problem for isotropic elastodynamics
HTML articles powered by AMS MathViewer
- by Lizabeth V. Rachele
- Trans. Amer. Math. Soc. 355 (2003), 4781-4806
- DOI: https://doi.org/10.1090/S0002-9947-03-03268-9
- Published electronically: July 28, 2003
- PDF | Request permission
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.References
- H.-W. Knobloch and B. Aulbach, The role of center manifolds in ordinary differential equations, Equadiff 5 (Bratislava, 1981) Teubner-Texte zur Mathematik, vol. 47, Teubner, Leipzig, 1982, pp. 179–189. MR 715971
- Yu. V. Egorov, Linear differential equations of principal type, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986. Translated from the Russian by Dang Prem Kumar. MR 872855
- Alain Grigis and Johannes Sjöstrand, Microlocal analysis for differential operators, London Mathematical Society Lecture Note Series, vol. 196, Cambridge University Press, Cambridge, 1994. An introduction. MR 1269107, DOI 10.1017/CBO9780511721441
- Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035, DOI 10.1007/978-3-642-96750-4
- Masaru Ikehata, Gen Nakamura, and Masahiro Yamamoto, Uniqueness in inverse problems for the isotropic Lamé system, J. Math. Sci. Univ. Tokyo 5 (1998), no. 4, 627–692. MR 1675236
- M. H. Protter, Unique continuation for elliptic equations, Trans. Amer. Math. Soc. 95 (1960), 81–91. MR 113030, DOI 10.1090/S0002-9947-1960-0113030-3
- Lizabeth V. Rachele, An inverse problem in elastodynamics: uniqueness of the wave speeds in the interior, J. Differential Equations 162 (2000), no. 2, 300–325. MR 1751708, DOI 10.1006/jdeq.1999.3657
- Lizabeth V. Rachele, Boundary determination for an inverse problem in elastodynamics, Comm. Partial Differential Equations 25 (2000), no. 11-12, 1951–1996. MR 1789918, DOI 10.1080/03605300008821575
- L. Rachele. Uniqueness in inverse problems for elastic media with residual stress. To appear in Comm. Partial Diff. Eq.
- Rakesh and William W. Symes, Uniqueness for an inverse problem for the wave equation, Comm. Partial Differential Equations 13 (1988), no. 1, 87–96. MR 914815, DOI 10.1080/03605308808820539
- V. A. Sharafutdinov, Integral geometry of tensor fields, Inverse and Ill-posed Problems Series, VSP, Utrecht, 1994. MR 1374572, DOI 10.1515/9783110900095
- V. A. Sharafutdinov, Integral geometry of a tensor field on a manifold with upper-bounded curvature, Sibirsk. Mat. Zh. 33 (1992), no. 3, 192–204, 221 (Russian, with Russian summary); English transl., Siberian Math. J. 33 (1992), no. 3, 524–533 (1993). MR 1178471, DOI 10.1007/BF00970902
- John Sylvester and Gunther Uhlmann, Inverse problems in anisotropic media, Inverse scattering and applications (Amherst, MA, 1990) Contemp. Math., vol. 122, Amer. Math. Soc., Providence, RI, 1991, pp. 105–117. MR 1135861, DOI 10.1090/conm/122/1135861
- François Trèves, Introduction to pseudodifferential and Fourier integral operators. Vol. 2, University Series in Mathematics, Plenum Press, New York-London, 1980. Fourier integral operators. MR 597145
- Gunther Uhlmann, Inverse boundary value problems and applications, Astérisque 207 (1992), 6, 153–211. Méthodes semi-classiques, Vol. 1 (Nantes, 1991). MR 1205179
Bibliographic 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
- 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)
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 4781-4806
- MSC (2000): Primary 35R30
- DOI: https://doi.org/10.1090/S0002-9947-03-03268-9
- MathSciNet review: 1997584