UNIQUENESS OF THE INVERSE TRANSMISSION SCATTERING WITH A CONDUCTIVE BOUNDARY CONDITION

Jianli XIANG; Guozheng YAN

doi:10.1007/s10473-021-0318-7

Acta mathematica scientia, Series B >

2021 , Vol. 41 >Issue 3: 925 - 940

DOI: https://doi.org/10.1007/s10473-021-0318-7

Articles

UNIQUENESS OF THE INVERSE TRANSMISSION SCATTERING WITH A CONDUCTIVE BOUNDARY CONDITION

Jianli XIANG ,
Guozheng YAN

Expand

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

Jianli XIANG,E-mail:xiangjl@mails.ccnu.edu.cn

Received date: 2020-03-05

Revised date: 2020-05-25

Online published: 2021-06-07

Supported by

This research is supported by NSFC (11571132).

Fold

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 $L^{p}(\partial\Omega)$ ($1

<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.

Key words： Acoustic wave; uniqueness; mixed reciprocity relation; modified interior transmission problem; a priori estimates

Cite this article

Jianli XIANG , Guozheng YAN . UNIQUENESS OF THE INVERSE TRANSMISSION SCATTERING WITH A CONDUCTIVE BOUNDARY CONDITION[J]. Acta mathematica scientia, Series B, 2021 , 41(3) : 925 -940 . DOI: 10.1007/s10473-021-0318-7

References

[1] Angell T S, Kirsch A. The conductive boundary condition for Maxwell's equation. SIAM J Appl Math, 1992, 52(6):1597-1610
[2] 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
[3] 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
[4] Bondarenko O, Liu X D. The factorization method for inverse obstacle scattering with conductive boundary condition. Inverse Probl, 2013, 29(9):095021
[5] Cakoni F, Colton D. A Qualitative Approach to Inverse Scattering Theory. Berlin:Springer, 2014
[6] Cakoni F, Colton D, Haddar H. Inverse Scattering Theory and Transmission Eigenvalues. Inverse Scattering Theory, 2016
[7] Cakoni F, Colton D, Haddar H. The linear sampling method for anisotropic media. J Comput Appl Math, 2002, 146(2):285-299
[8] Colton D, Kress R. Integral Equation Methods in Scattering Theory. New York:Wiley, 1983
[9] Colton D, Kress R. Inverse Acoustic and Electromagnetic Scattering Theory. 4th ed. Springer Nature Switzerland AG, 2019
[10] Colton D, Kress R. Using fundamental solutions in inverse scattering. Inverse Probl, 2006, 22(3):49-66
[11] Colton D, Kress R, Monk P. Inverse scattering from an orthotropic medium. J Comput Appl Math, 1997, 81(2):269-298
[12] Gerlach T, Kress R. Uniqueness in inverse obstacle scattering with conductive boundary condition. Inverse Probl, 1996, 12(5):619-625
[13] Gintides D. Local uniqueness for the inverse scattering problem in acoustics via the Faber-Krahn inequality. Inverse Probl, 2005, 21(4):1195-1205
[14] Hähner P. On the uniqueness of the shape of a penetrable, anisotropic obstacle. J Comput Appl Math, 2000, 116(1):167-180
[15] Harris I, Kleefeld A. The inverse scattering problem for a conductive boundary condition and transmission eigenvalues. Appl Anal, 2020, 99(3):508-529
[16] Hettlich F. On the uniqueness of the inverse conductive scattering problem for the Helmholtz equation. Inverse Probl, 1994, 10(1):129-144
[17] Isakov V. On uniqueness in the inverse transmission scattering problem. Commun Part Diff Equ, 1990, 15(11):1565-1587
[18] Kirsch A, Kress R. Uniqueness in inverse obstacle scattering. Inverse Probl, 1993, 9(2):285-299
[19] Kress R. Uniqueness and numerical methods in inverse obstacle scattering. J Phys Conf Ser, 2007, 73(1):012003
[20] Lax P D, Phillips R S. Scattoing Theory. New York Academic, 1967
[21] 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
[22] 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
[23] 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
[24] 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
[25] Piana M. On uniqueness for anisotropic inhomogeneous inverse scattering problems. Inverse Probl, 1998, 14(6):1565-1579
[26] Potthast R. A point-source method for inverse acoustic and electromagnetic obstacle scattering problems. IMA J Appl Math, 1998, 61(2):119-140
[27] Potthast R. On the convergence of a new Newton-type method in inverse scattering. Inverse Probl, 2001, 17(5):1419-1434
[28] Potthast R. Point sources and multipoles in inverse scattering theory. Chapman and Hall/CRC, 2001
[29] Qu F L, Yang J Q. On recovery of an inhomogeneous cavity in inverse acoustic scattering. Inverse Probl Imag, 2018, 12(2):281-291
[30] 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
[31] Ramm A G. New method for proving uniqueness theorems for obstacle inverse scattering problems. Appl Math Lett, 1993, 6(6):19-21
[32] 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
[33] Valdivia N. Uniqueness in inverse obstacle scattering with conductive boundary conditions. Appl Anal, 2004, 83(8):825-851
[34] 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

Options

Abstract

Outlines

模态框（Modal）标题

Abstract

Cite this article

References