Abstract
This paper considers the inverse acoustic wave scattering by a bounded penetrable obstacle with a conductive boundary condition. We will show that the penetrable scatterer can be uniquely determined by its far-field pattern of the scattered field for all incident plane waves at a fixed wave number. In the first part of this paper, adequate preparations for the main uniqueness result are made. We establish the mixed reciprocity relation between the far-field pattern corresponding to point sources and the scattered field corresponding to plane waves. Then the well-posedness of a modified interior transmission problem is deeply investigated by the variational method. Finally, the a priori estimates of solutions to the general transmission problem with boundary data in Lp (∂Ω) (1 < p < 2) are proven by the boundary integral equation method. In the second part of this paper, we give a novel proof on the uniqueness of the inverse conductive scattering problem.
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References
Angell T S, Kirsch A. The conductive boundary condition for Maxwell’s equation. SIAM J Appl Math, 1992, 52(6): 1597–1610
Angell T S, Kleinman R E, Hettlich F. The resistive and conductive problems for the exterior Helmholtz Equation. SIAM J Appl Math, 1990, 50(6): 1607–1622
Bondarenko O, Harris I, Kleefeld A. The interior transmission eigenvalue problem for an inhomogeneous media with a conductive boundary. Appl Anal, 2017, 96(1): 2–22
Bondarenko O, Liu X D. The factorization method for inverse obstacle scattering with conductive boundary condition. Inverse Probl, 2013, 29(9): 095021
Cakoni F, Colton D. A Qualitative Approach to Inverse Scattering Theory. Berlin: Springer, 2014
Cakoni F, Colton D, Haddar H. Inverse Scattering Theory and Transmission Eigenvalues. Inverse Scattering Theory, 2016
Cakoni F, Colton D, Haddar H. The linear sampling method for anisotropic media. J Comput Appl Math, 2002, 146(2): 285–299
Colton D, Kress R. Integral Equation Methods in Scattering Theory. New York: Wiley, 1983
Colton D, Kress R. Inverse Acoustic and Electromagnetic Scattering Theory. 4th ed. Springer Nature Switzerland AG, 2019
Colton D, Kress R. Using fundamental solutions in inverse scattering. Inverse Probl, 2006, 22(3): 49–66
Colton D, Kress R, Monk P. Inverse scattering from an orthotropic medium. J Comput Appl Math, 1997, 81(2): 269–298
Gerlach T, Kress R. Uniqueness in inverse obstacle scattering with conductive boundary condition. Inverse Probl, 1996, 12(5): 619–625
Gintides D. Local uniqueness for the inverse scattering problem in acoustics via the Faber-Krahn inequality. Inverse Probl, 2005, 21(4): 1195–1205
Hähner P. On the uniqueness of the shape of a penetrable, anisotropic obstacle. J Comput Appl Math, 2000, 116(1): 167–180
Harris I, Kleefeld A. The inverse scattering problem for a conductive boundary condition and transmission eigenvalues. Appl Anal, 2020, 99(3): 508–529
Hettlich F. On the uniqueness of the inverse conductive scattering problem for the Helmholtz equation. Inverse Probl, 1994, 10(1): 129–144
Isakov V. On uniqueness in the inverse transmission scattering problem. Commun Part Diff Equ, 1990, 15(11): 1565–1587
Kirsch A, Kress R. Uniqueness in inverse obstacle scattering. Inverse Probl, 1993, 9(2): 285–299
Kress R. Uniqueness and numerical methods in inverse obstacle scattering. J Phys Conf Ser, 2007, 73(1): 012003
Lax P D, Phillips R S. Scattoing Theory. New York Academic, 1967
Liu X D, Zhang B. Direct and inverse obstacle scattering problems in a piecewise homogeneous medium. SIAM J Appl Math, 2010, 70(8): 3105–3120
Liu X D, Zhang B. Inverse scattering by an inhomogeneous penetrable obstacle in a piecewise homogeneous medium. Acta Mathematica Scientia, 2012, 32B(4): 1281–1297
Liu X D, Zhang B, Hu G H. Uniqueness in the inverse scattering problem in a piecewise homogeneous medium. Inverse Probl, 2010, 26(1): 015002
Mitrea D, Mitrea M. Uniqueness for inverse conductivity and transmission problems in the class of Lipschitz domains. Commun Part Diff Equ, 1998, 23(7): 1419–1448
Piana M. On uniqueness for anisotropic inhomogeneous inverse scattering problems. Inverse Probl, 1998, 14(6): 1565–1579
Potthast R. A point-source method for inverse acoustic and electromagnetic obstacle scattering problems. IMA J Appl Math, 1998, 61(2): 119–140
Potthast R. On the convergence of a new Newton-type method in inverse scattering. Inverse Probl, 2001, 17(5): 1419–1434
Potthast R. Point sources and multipoles in inverse scattering theory. Chapman and Hall/CRC, 2001
Qu F L, Yang J Q. On recovery of an inhomogeneous cavity in inverse acoustic scattering. Inverse Probl Imag, 2018, 12(2): 281–291
Qu F L, Yang J Q, Zhang B. Recovering an elastic obstacle containing embedded objects by the acoustic far-field measurements. Inverse Probl, 2018, 34(1): 015002
Ramm A G. New method for proving uniqueness theorems for obstacle inverse scattering problems. Appl Math Lett, 1993, 6(6): 19–21
Stefanov P, Uhlmann G. Local uniqueness for the fixed energy fixed angle inverse problem in obstacle scattering. P Amer Math Soc, 2004, 132(5): 1351–1354
Valdivia N. Uniqueness in inverse obstacle scattering with conductive boundary conditions. Appl Anal, 2004, 83(8): 825–851
Yang J Q, Zhang B, Zhang H W. Uniqueness in inverse acoustic and electromagnetic scattering by penetrable obstacles with embedded objects. J Diff Equ, 2018, 265(12): 6352–6383
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This research is supported by NSFC (11571132).
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Xiang, J., Yan, G. Uniqueness of the Inverse Transmission Scattering with a Conductive Boundary Condition. Acta Math Sci 41, 925–940 (2021). https://doi.org/10.1007/s10473-021-0318-7
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DOI: https://doi.org/10.1007/s10473-021-0318-7
Key words
- Acoustic wave
- uniqueness
- mixed reciprocity relation
- modified interior transmission problem
- a priori estimates