Uniqueness of the density in an inverse problem for isotropic elastodynamics
Author:
Lizabeth V. Rachele
Journal:
Trans. Amer. Math. Soc. 355 (2003), 47814806
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, 3dimensional 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 DirichlettoNeumann 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 DirichlettoNeumann map. Here we extend that result by showing that the density is also determined in the interior by the DirichlettoNeumann 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 threedimensional problem.
 [De]
H.W.
Knobloch and B.
Aulbach, The role of center manifolds in ordinary differential
equations, Equadiff 5 (Bratislava, 1981) TeubnerTexte zur Math.,
vol. 47, Teubner, Leipzig, 1982, pp. 179–189. MR 715971
(84k:58126)
 [Eg]
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
(88c:35003)
 [GrSj]
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 (95d:35009)
 [Hö]
Lars
Hörmander, The analysis of linear partial differential
operators. I, Grundlehren der Mathematischen Wissenschaften
[Fundamental Principles of Mathematical Sciences], vol. 256,
SpringerVerlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035
(85g:35002a)
 [INY]
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
(2000i:35213)
 [Pr]
M.
H. Protter, Unique continuation for elliptic
equations, Trans. Amer. Math. Soc. 95 (1960), 81–91. MR 0113030
(22 #3871), http://dx.doi.org/10.1090/S00029947196001130303
 [R I]
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
(2001e:35177), http://dx.doi.org/10.1006/jdeq.1999.3657
 [R II]
Lizabeth
V. Rachele, Boundary determination for an inverse problem in
elastodynamics, Comm. Partial Differential Equations
25 (2000), no. 1112, 1951–1996. MR 1789918
(2001m:35322), http://dx.doi.org/10.1080/03605300008821575
 [R III]
L. Rachele. Uniqueness in inverse problems for elastic media with residual stress. To appear in Comm. Partial Diff. Eq.
 [RaSy]
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
(89f:35208), http://dx.doi.org/10.1080/03605308808820539
 [Sh I]
V.
A. Sharafutdinov, Integral geometry of tensor fields, Inverse
and Illposed Problems Series, VSP, Utrecht, 1994. MR 1374572
(97h:53077)
 [Sh II]
V.
A. Sharafutdinov, Integral geometry of a tensor field on a manifold
with upperbounded 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
(94d:53116), http://dx.doi.org/10.1007/BF00970902
 [SU]
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
(92k:35289), http://dx.doi.org/10.1090/conm/122/1135861
 [Tr]
François
Trèves, Introduction to pseudodifferential and Fourier
integral operators. Vol. 2, Plenum Press, New YorkLondon, 1980.
Fourier integral operators; The University Series in Mathematics. MR 597145
(82i:58068)
 [U]
Gunther
Uhlmann, Inverse boundary value problems and applications,
Astérisque 207 (1992), 6, 153–211.
Méthodes semiclassiques, Vol.\ 1 (Nantes, 1991). MR 1205179
(94e:35146)
 [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
 [GrSj]
 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, SpringerVerlag, Berlin, 1983. MR 85g:35002a
 [INY]
 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 (1112), 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.
 [RaSy]
 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 IllPosed Problems Series. VSP, Utrecht, 1994. MR 97h:53077
 [Sh II]
 V. Sharafutdinov. Integral geometry of a tensor field on a manifold with upperbounded curvature, Siberian Math. J. 33 (3), 1992, 524533. MR 94d:53116
 [SU]
 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 semiclassiques, 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:
http://dx.doi.org/10.1090/S0002994703032689
PII:
S 00029947(03)032689
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
