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

Rondi, Luca

doi:10.1090/S0002-9947-02-03105-7

Uniqueness for the determination of sound-soft defects in an inhomogeneous planar medium by acoustic boundary measurements
HTML articles powered by AMS MathViewer

by Luca Rondi PDF

Trans. Amer. Math. Soc. 355 (2003), 213-239 Request permission

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

David R. Adams and Lars Inge Hedberg, Function spaces and potential theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314, Springer-Verlag, Berlin, 1996. MR 1411441, DOI 10.1007/978-3-662-03282-4
Giovanni Alessandrini and Alvaro Diaz Valenzuela, Unique determination of multiple cracks by two measurements, SIAM J. Control Optim. 34 (1996), no. 3, 913–921. MR 1384959, DOI 10.1137/S0363012994262853
G. Alessandrini and E. DiBenedetto, Determining $2$-dimensional cracks in $3$-dimensional bodies: uniqueness and stability, Indiana Univ. Math. J. 46 (1997), no. 1, 1–82. MR 1462795, DOI 10.1512/iumj.1997.46.1332
G. Alessandrini and R. Magnanini, Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions, SIAM J. Math. Anal. 25 (1994), no. 5, 1259–1268. MR 1289138, DOI 10.1137/S0036141093249080
Lipman Bers, Fritz John, and Martin Schechter, Partial differential equations, Lectures in Applied Mathematics, Vol. III, Interscience Publishers, a division of John Wiley & Sons, Inc., New York-London-Sydney, 1964. With special lectures by Lars Garding and A. N. Milgram. MR 0163043
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
David Colton, Joe Coyle, and Peter Monk, Recent developments in inverse acoustic scattering theory, SIAM Rev. 42 (2000), no. 3, 369–414. MR 1786932, DOI 10.1137/S0036144500367337
David Colton and Rainer Kress, Inverse acoustic and electromagnetic scattering theory, 2nd ed., Applied Mathematical Sciences, vol. 93, Springer-Verlag, Berlin, 1998. MR 1635980, DOI 10.1007/978-3-662-03537-5
Avner Friedman and Michael Vogelius, Determining cracks by boundary measurements, Indiana Univ. Math. J. 38 (1989), no. 3, 527–556. MR 1017323, DOI 10.1512/iumj.1989.38.38025
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1207810
Victor Isakov, On uniqueness in the inverse transmission scattering problem, Comm. Partial Differential Equations 15 (1990), no. 11, 1565–1587. MR 1079603, DOI 10.1080/03605309908820737
Victor Isakov, Inverse problems for partial differential equations, Applied Mathematical Sciences, vol. 127, Springer-Verlag, New York, 1998. MR 1482521, DOI 10.1007/978-1-4899-0030-2
Andreas Kirsch and Lassi Päivärinta, On recovering obstacles inside inhomogeneities, Math. Methods Appl. Sci. 21 (1998), no. 7, 619–651. MR 1615992, DOI 10.1002/(SICI)1099-1476(19980510)21:7<619::AID-MMA940>3.3.CO;2-G
Rainer Kress, Inverse scattering from an open arc, Math. Methods Appl. Sci. 18 (1995), no. 4, 267–293. MR 1319999, DOI 10.1002/mma.1670180403
J. R. Kuttler and V. G. Sigillito, Eigenvalues of the Laplacian in two dimensions, SIAM Rev. 26 (1984), no. 2, 163–193. MR 738929, DOI 10.1137/1026033
W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 43–77. MR 161019
Norman G. Meyers, An $L^{p}$e-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 189–206. MR 159110
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/).
Guido Stampacchia, Problemi al contorno ellitici, con dati discontinui, dotati di soluzionie hölderiane, Ann. Mat. Pura Appl. (4) 51 (1960), 1–37 (Italian). MR 126601, DOI 10.1007/BF02410941
M. Tsuji, Potential theory in modern function theory, Chelsea Publishing Co., New York, 1975. Reprinting of the 1959 original. MR 0414898
William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3