Uniqueness for the determination of soundsoft defects in an inhomogeneous planar medium by acoustic boundary measurements
Author:
Luca Rondi
Journal:
Trans. Amer. Math. Soc. 355 (2003), 213239
MSC (2000):
Primary 35R30
Published electronically:
September 5, 2002
MathSciNet review:
1928086
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Abstract: We consider the inverse problem of determining shape and location of soundsoft defects inside a known planar inhomogeneous and anisotropic medium through acoustic imaging at low frequency. In order to determine the defects, we perform acoustic boundary measurements, with prescribed boundary conditions of different types. We prove that at most two, suitably chosen, measurements allow us to uniquely determine multiple defects under minimal regularity assumptions on the defects and the medium containing them. Finally, we treat applications of these results to the case of inverse scattering.
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V. Isakov, On uniqueness in the inverse transmission scattering problem, Comm. Partial Differential Equations 15 (1990), pp. 15651587. MR 91i:35203
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V. Isakov, Inverse Problems for Partial Differential Equations, SpringerVerlag, New York, 1998. MR 99b:35211
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A. Kirsch and L. Päivärinta, On recovering obstacles inside inhomogeneities, Math. Meth. Appl. Sci. 21 (1998), pp. 619651. MR 99b:35214
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R. Kress, Inverse scattering from an open arc, Math. Meth. Appl. Sci. 18 (1995), pp. 267293. MR 95k:35219
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W. Littman, G. Stampacchia and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1963), pp. 4377. MR 28:4228
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N. G. Meyers, An estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1963), pp. 189206. MR 28:2328
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L. Rondi, Uniqueness and Optimal Stability for the Determination of Multiple Defects by Electrostatic Measurements, Ph.D. thesis, S.I.S.S.A.I.S.A.S., Trieste, 1999 (downloadable from http://www.sissa.it/library/).
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G. Stampacchia, Problemi al contorno ellittici, con dati discontinui, dotati di soluzioni hölderiane, Ann. Mat. Pura Appl. 51 (1960), pp. 137. MR 23:A3897
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M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publishing Company, New York, 1975, Reprint of the 1959 original. MR 54:2990
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W. P. Ziemer, Weakly Differentiable Functions, SpringerVerlag, New York, 1989. MR 91e:46046
 1.
 D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, SpringerVerlag, Berlin, 1996. MR 97j:46024
 2.
 G. Alessandrini and A. Diaz Valenzuela, Unique determination of multiple cracks by two measurements, SIAM J. Control Optim. 34 (1996), pp. 913921. MR 97a:78017
 3.
 G. Alessandrini and E. Di Benedetto, Determining 2dimensional cracks in 3dimensional bodies: uniqueness and stability, Indiana Univ. Math. J. 46 (1997), pp. 182. MR 98g:35207
 4.
 G. Alessandrini and R. Magnanini, Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions, SIAM J. Math. Anal. 25 (1994), pp. 12591268. MR 95f:35180
 5.
 L. Bers, F. John and M. Schechter, Partial Differential Equations, Interscience Publishers, New York, London, Sidney, 1964. MR 29:346
 6.
 L. Bers and L. Nirenberg, On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications, in Convegno Internazionale sulle Equazioni Lineari alle Derivate Parziali, Trieste, Edizioni Cremonese, Roma, 1955, pp. 111140. MR 17:974d
 7.
 D. Colton, J. Coyle and P. Monk, Recent developments in inverse acoustic scattering theory, SIAM Review 42 (2000), pp. 369414. MR 2001f:76066
 8.
 D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Second edition, SpringerVerlag, Berlin, 1998. MR 99c:35181
 9.
 A. Friedman and M. Vogelius, Determining cracks by boundary measurements, Indiana Univ. Math. J. 38 (1989), pp. 527556. MR 91b:35109
 10.
 D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, SpringerVerlag, Berlin, 1983. MR 86c:35035
 11.
 P. Hartman and A. Wintner, On the local behavior of solutions of nonparabolic partial differential equations, Amer. J. Math. 75 (1953), pp. 449476. MR 15:318b
 12.
 J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford Univ. Press, New York, 1993. MR 94e:31003
 13.
 V. Isakov, On uniqueness in the inverse transmission scattering problem, Comm. Partial Differential Equations 15 (1990), pp. 15651587. MR 91i:35203
 14.
 V. Isakov, Inverse Problems for Partial Differential Equations, SpringerVerlag, New York, 1998. MR 99b:35211
 15.
 A. Kirsch and L. Päivärinta, On recovering obstacles inside inhomogeneities, Math. Meth. Appl. Sci. 21 (1998), pp. 619651. MR 99b:35214
 16.
 R. Kress, Inverse scattering from an open arc, Math. Meth. Appl. Sci. 18 (1995), pp. 267293. MR 95k:35219
 17.
 J. R. Kuttler and V. G. Sigillito, Eigenvalues of the Laplacian in two dimensions, SIAM Review 26 (1984), pp. 163193. MR 85k:65086
 18.
 W. Littman, G. Stampacchia and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1963), pp. 4377. MR 28:4228
 19.
 N. G. Meyers, An estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1963), pp. 189206. MR 28:2328
 20.
 L. Rondi, Uniqueness and Optimal Stability for the Determination of Multiple Defects by Electrostatic Measurements, Ph.D. thesis, S.I.S.S.A.I.S.A.S., Trieste, 1999 (downloadable from http://www.sissa.it/library/).
 21.
 G. Stampacchia, Problemi al contorno ellittici, con dati discontinui, dotati di soluzioni hölderiane, Ann. Mat. Pura Appl. 51 (1960), pp. 137. MR 23:A3897
 22.
 M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publishing Company, New York, 1975, Reprint of the 1959 original. MR 54:2990
 23.
 W. P. Ziemer, Weakly Differentiable Functions, SpringerVerlag, New York, 1989. MR 91e:46046
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Additional Information
Luca Rondi
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Address at time of publication:
Dipartimento di Scienze Matematiche, Università degli Studi di Trieste, Trieste, Italy
Email:
rondi@mathsun1.univ.trieste.it
DOI:
http://dx.doi.org/10.1090/S0002994702031057
PII:
S 00029947(02)031057
Received by editor(s):
November 12, 2001
Received by editor(s) in revised form:
March 19, 2002
Published electronically:
September 5, 2002
Article copyright:
© Copyright 2002
American Mathematical Society
