Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Uniqueness for the determination of sound-soft defects in an inhomogeneous planar medium by acoustic boundary measurements

Author: Luca Rondi
Journal: Trans. Amer. Math. Soc. 355 (2003), 213-239
MSC (2000): Primary 35R30
Published electronically: September 5, 2002
MathSciNet review: 1928086
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the inverse problem of determining shape and location of sound-soft 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.

References [Enhancements On Off] (What's this?)

  • 1. D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, 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. 913-921. MR 97a:78017
  • 3. G. Alessandrini and E. Di Benedetto, Determining 2-dimensional cracks in 3-dimensional bodies: uniqueness and stability, Indiana Univ. Math. J. 46 (1997), pp. 1-82. 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. 1259-1268. 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. 111-140. MR 17:974d
  • 7. D. Colton, J. Coyle and P. Monk, Recent developments in inverse acoustic scattering theory, SIAM Review 42 (2000), pp. 369-414. MR 2001f:76066
  • 8. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Second edition, Springer-Verlag, Berlin, 1998. MR 99c:35181
  • 9. A. Friedman and M. Vogelius, Determining cracks by boundary measurements, Indiana Univ. Math. J. 38 (1989), pp. 527-556. MR 91b:35109
  • 10. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, Springer-Verlag, Berlin, 1983. MR 86c:35035
  • 11. P. Hartman and A. Wintner, On the local behavior of solutions of non-parabolic partial differential equations, Amer. J. Math. 75 (1953), pp. 449-476. 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. 1565-1587. MR 91i:35203
  • 14. V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, New York, 1998. MR 99b:35211
  • 15. A. Kirsch and L. Päivärinta, On recovering obstacles inside inhomogeneities, Math. Meth. Appl. Sci. 21 (1998), pp. 619-651. MR 99b:35214
  • 16. R. Kress, Inverse scattering from an open arc, Math. Meth. Appl. Sci. 18 (1995), pp. 267-293. MR 95k:35219
  • 17. J. R. Kuttler and V. G. Sigillito, Eigenvalues of the Laplacian in two dimensions, SIAM Review 26 (1984), pp. 163-193. 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. 43-77. MR 28:4228
  • 19. N. G. Meyers, An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1963), pp. 189-206. 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
  • 21. G. Stampacchia, Problemi al contorno ellittici, con dati discontinui, dotati di soluzioni hölderiane, Ann. Mat. Pura Appl. 51 (1960), pp. 1-37. 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, Springer-Verlag, New York, 1989. MR 91e:46046

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35R30

Retrieve articles in all journals with MSC (2000): 35R30

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

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

American Mathematical Society