June  2021, 15(3): 539-554. doi: 10.3934/ipi.2021004

The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation

1. 

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

* Corresponding author: Guozheng Yan

Received  June 2020 Revised  October 2020 Published  June 2021 Early access  December 2020

Fund Project: The first author is supported by the Innovative Funding Project from Central China Normal University (Grant No. 2019CXZZ078). The second author is supported by the National Natural Science Foundation of China (Grant No. 11571132)

This paper considers the inverse elastic wave scattering by a bounded penetrable or impenetrable scatterer. We propose a novel technique to show that the elastic obstacle can be uniquely determined by its far-field pattern associated with all incident plane waves at a fixed frequency. In the first part of this paper, we establish the mixed reciprocity relation between the far-field pattern corresponding to special point sources and the scattered field corresponding to plane waves, and the mixed reciprocity relation is the key point to show the uniqueness results. In the second part, besides the mixed reciprocity relation, a priori estimates of solution to the transmission problem with boundary data in $ [L^{\mathrm{q}}(\partial\Omega)]^{3} $ ($ 1<\mathrm{q}<2 $) is deeply investigated by the integral equation method and also we have constructed a well-posed modified static interior transmission problem on a small domain to obtain the uniqueness result.

Citation: Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, 2021, 15 (3) : 539-554. doi: 10.3934/ipi.2021004
References:
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A. Adams and J. F. Fournier, Sobolev Spaces, 2$^{nd}$ edition, Elsevier, Singapore, 2003.  Google Scholar

[2]

J. F. Ahner and G. C. Hsiao, A Neumann series representation for solutions to boundary value problems in dynamic elasticity, Quart. Appl. Math., 33 (1975/76), 73-80.  doi: 10.1090/qam/449124.  Google Scholar

[3]

J. F. Ahner and G. C. Hsiao, On the two-dimensional exterior boundary-value problems of elasticity, Siam J. Appl. Math., 31 (1976), 677-685.  doi: 10.1137/0131060.  Google Scholar

[4]

K. A. Anagnostopoulos and A. Charalambopoulos, The linear sampling method for the transmission problem in 2D anisotropic elasticity, Inverse Problems, 22 (2006), 553-577.  doi: 10.1088/0266-5611/22/2/011.  Google Scholar

[5]

T. Arens, Linear sampling methods for 2D inverse elastic wave scattering, Inverse Problems, 17 (2001), 1445-1464.  doi: 10.1088/0266-5611/17/5/314.  Google Scholar

[6]

A. Charalambopoulos, On the interior transmission problem in nondissipative, inhomogeneous, anisotropic elasticity, J. Elasticity, 67 (2002), 149-170.  doi: 10.1023/A:1023958030304.  Google Scholar

[7]

A. Charalambopoulos and K. A. Anagnostopoulos, On the spectrum of the interior transmission problem in isotropic elasticity, J. Elasticity, 90 (2008), 295-313.  doi: 10.1007/s10659-007-9146-9.  Google Scholar

[8]

A. CharalambopoulosA. KirschK. A. AnagnostopoulosD. Gintides and K. Kiriaki, The factorization method in inverse elastic scattering from penetrable bodies, Inverse Problems, 23 (2006), 27-51.  doi: 10.1088/0266-5611/23/1/002.  Google Scholar

[9]

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983.  Google Scholar

[10]

D. Colton and R. Kress, Using fundamental solutions in inverse scattering, Inverse Problems, 22 (2006), 49-66.  doi: 10.1088/0266-5611/22/3/R01.  Google Scholar

[11]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4$^{th}$ edition, Springer Nature Switzerland AG, 2019. doi: 10.1007/978-3-030-30351-8.  Google Scholar

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D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering, IMA J. Appl. Math., 31 (1983), 253-259.  doi: 10.1093/imamat/31.3.253.  Google Scholar

[13]

H. A. Diao, H. Y. Liu and L. Wang, On generalized Holmgren's principle to the Lamé operator with applications to inverse elastic problems, Calculus of Variations and Partial Differential Equations, in press, 59 (2020), Paper No. 179, 50 pp. doi: 10.1007/s00526-020-01830-5.  Google Scholar

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J. Elschner and M. Yamamoto, Uniqueness in inverse elastic scattering with finitely many incident waves, Inverse Problems, 26 (2010), 045005, 8pp. doi: 10.1088/0266-5611/26/4/045005.  Google Scholar

[15]

T. Gerlach and R. Kress, Uniqueness in inverse obstacle scattering with conductive boundary condition, Inverse Problems, 12 (1996), 619-625.  doi: 10.1088/0266-5611/12/5/006.  Google Scholar

[16]

D. Gintides, Local uniqueness for the inverse scattering problem in acoustics via the Faber-Krahn inequality, Inverse Problems, 21 (2005), 1195-1205.  doi: 10.1088/0266-5611/21/4/001.  Google Scholar

[17]

D. Gintides and L. Midrinos, Inverse scattering problem for a rigid scatterer or a cavity in elastodynamics, Zamm. J. Appl. Math. Mech., 91 (2011), 276-287.  doi: 10.1002/zamm.201000098.  Google Scholar

[18]

D. Gintides and M. Sini, Identification of obstacles using only the scatteres P-waves or the scattered S-waves, Inverse Probl. Imaging, 6 (2012), 39-55.  doi: 10.3934/ipi.2012.6.39.  Google Scholar

[19]

D. GintidesM. Sini and N. T. Thành, Detection of point-like scatterers using one type of scattered elastic waves, J. Comput. Appl. Math., 236 (2012), 2137-2145.  doi: 10.1016/j.cam.2011.09.036.  Google Scholar

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer, New York, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[21]

P. Hähner and G. C. Hsiao, Uniqueness theorems in inverse obstacle scattering of elastic waves, Inverse Problems, 9 (1993), 525-534.  doi: 10.1088/0266-5611/9/5/002.  Google Scholar

[22]

G. H. Hu, A. Kirsch and M. Sini, Some inverse problems arising from elastic scattering by rigid obstacles, Inverse Problems, 29 (2013), 015009, 21pp. doi: 10.1088/0266-5611/29/1/015009.  Google Scholar

[23]

G. H. HuJ. Z. Li and H. Y. Liu, Recovering complex elastic scatterers by a single far-field pattern, J. Differential Equations, 257 (2014), 469-489.  doi: 10.1016/j.jde.2014.04.007.  Google Scholar

[24]

M. Kar and M. Sini, On the inverse elastic scattering by interfaces using one type of scattered waves, J. Elasticity, 118 (2015), 15-38.  doi: 10.1007/s10659-014-9474-5.  Google Scholar

[25]

A. Kirsch and R. Kress, Uniqueness in inverse obstacle scattering, Inverse Problems, 9 (1993), 285-299.  doi: 10.1088/0266-5611/9/2/009.  Google Scholar

[26]

R. Kress, Uniqueness and numerical methods in inverse obstacle scattering, J. Physics: Conference Series, 73 (2007), 012003. doi: 10.1088/1742-6596/73/1/012003.  Google Scholar

[27]

V. D. Kupradze, Potential Methods in the Theory of Elasticity, Jerusalem: Israeli Program for Scientific Translations, 1965.  Google Scholar

[28]

V. D. Kupradze, Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, Amsterdam: North-Holland, 1979. Google Scholar

[29]

J. J. LaiH. Y. LiuJ. N. Xiao and Y. F. Xu, The decoupling of elastic waves from a weak formulation perspective, East Asian Journal on Applied Mathematics, 9 (2019), 241-251.  doi: 10.4208/eajam.080818.121018.  Google Scholar

[30]

P. D. Lax and R. S. Phillips, Scattering Theory, Pure and Applied Mathematics, Vol. 26 Academic Press, New York-London, 1967.  Google Scholar

[31]

H. Y. Liu and J. N. Xiao, Decoupling elastic waves and its applications, J. Differential Equations, 263 (2017), 4442-4480.  doi: 10.1016/j.jde.2017.05.022.  Google Scholar

[32]

H. Y. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), 515-524.  doi: 10.1088/0266-5611/22/2/008.  Google Scholar

[33]

X. D. Liu and B. Zhang, Direct and inverse obstacle scattering problems in a piecewise homogeneous medium, SIAM J. Appl. Math., 70 (2010), 3105-3120.  doi: 10.1137/090777578.  Google Scholar

[34]

X. D. Liu and B. Zhang, Inverse scattering by an inhomogeneous penetrable obstacle in a piecewise homogeneous medium, Acta Math. Sci., 32B (2012), 1281-1297.  doi: 10.1016/S0252-9602(12)60099-X.  Google Scholar

[35]

X. D. Liu, B. Zhang and G. H. Hu, Uniqueness in the inverse scattering problem in a piecewise homogeneous medium, Inverse Problems, 26 (2010), 015002, 14pp. doi: 10.1088/0266-5611/26/1/015002.  Google Scholar

[36]

P. A. Martin, On the scattering of elastic waves by an elastic inclusion in two dimensions, Quar. J. Mech. Appl. Math., 43 (1990), 275-291.  doi: 10.1093/qjmam/43.3.275.  Google Scholar

[37]

R. Potthast, A point source method for inverse acoustic and electromagnetic obstacle scattering problems, IMA J. Appl. Math., 61 (1998), 119-140.  doi: 10.1093/imamat/61.2.119.  Google Scholar

[38]

R. Potthast, On the convergence of a new Newton-type method in inverse scattering, Inverse Problems, 17 (2001), 1419-1434.  doi: 10.1088/0266-5611/17/5/312.  Google Scholar

[39]

F. L. Qu, J. Q. Yang and B. Zhang, Recovering an elastic obstacle containing embedded objects by the acoustic far-field measurements, Inverse Problems, 34 (2018), 015002, 8pp. doi: 10.1088/1361-6420/aa9c26.  Google Scholar

[40]

A. G. Ramm, New method for proving uniqueness theorems for obstacle inverse scattering problems, Appl. Math. Lett., 6 (1993), 19-21.  doi: 10.1016/0893-9659(93)90071-T.  Google Scholar

[41]

A. G. Ramm, Research anouncement uniqueness theorems for inverse obstacle scattering problems in Lipschitz domains, Appl. Anal., 59 (1995), 337-383.  doi: 10.1080/00036819508840411.  Google Scholar

[42]

L. Rondi, E. Sincich and M. Sini, Stable determination of a rigid scatterer in elastodynamics, arXiv: 2007.06864v1. Google Scholar

[43]

P. Stefanov and G. Uhlmann, Local uniqueness for the fixed energy fixed angle inverse problem in obstacle scattering, Proc. Amer. Math. Soc, 132 (2004), 1351-1354.  doi: 10.1090/S0002-9939-03-07363-5.  Google Scholar

[44]

J. Q. YangB. Zhang and H. W. Zhang, Uniqueness in inverse acoustic and electromagnetic scattering by penetrable obstacles with embedded objects, J. Differential Equations, 265 (2018), 6352-6383.  doi: 10.1016/j.jde.2018.07.033.  Google Scholar

[45]

D. Y. Zhang and Y. K. Guo, Uniqueness results on phaseless inverse acoustic scattering with a reference ball, Inverse Problems, 34 (2018), 085002, 12pp. doi: 10.1088/1361-6420/aac53c.  Google Scholar

show all references

References:
[1]

A. Adams and J. F. Fournier, Sobolev Spaces, 2$^{nd}$ edition, Elsevier, Singapore, 2003.  Google Scholar

[2]

J. F. Ahner and G. C. Hsiao, A Neumann series representation for solutions to boundary value problems in dynamic elasticity, Quart. Appl. Math., 33 (1975/76), 73-80.  doi: 10.1090/qam/449124.  Google Scholar

[3]

J. F. Ahner and G. C. Hsiao, On the two-dimensional exterior boundary-value problems of elasticity, Siam J. Appl. Math., 31 (1976), 677-685.  doi: 10.1137/0131060.  Google Scholar

[4]

K. A. Anagnostopoulos and A. Charalambopoulos, The linear sampling method for the transmission problem in 2D anisotropic elasticity, Inverse Problems, 22 (2006), 553-577.  doi: 10.1088/0266-5611/22/2/011.  Google Scholar

[5]

T. Arens, Linear sampling methods for 2D inverse elastic wave scattering, Inverse Problems, 17 (2001), 1445-1464.  doi: 10.1088/0266-5611/17/5/314.  Google Scholar

[6]

A. Charalambopoulos, On the interior transmission problem in nondissipative, inhomogeneous, anisotropic elasticity, J. Elasticity, 67 (2002), 149-170.  doi: 10.1023/A:1023958030304.  Google Scholar

[7]

A. Charalambopoulos and K. A. Anagnostopoulos, On the spectrum of the interior transmission problem in isotropic elasticity, J. Elasticity, 90 (2008), 295-313.  doi: 10.1007/s10659-007-9146-9.  Google Scholar

[8]

A. CharalambopoulosA. KirschK. A. AnagnostopoulosD. Gintides and K. Kiriaki, The factorization method in inverse elastic scattering from penetrable bodies, Inverse Problems, 23 (2006), 27-51.  doi: 10.1088/0266-5611/23/1/002.  Google Scholar

[9]

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983.  Google Scholar

[10]

D. Colton and R. Kress, Using fundamental solutions in inverse scattering, Inverse Problems, 22 (2006), 49-66.  doi: 10.1088/0266-5611/22/3/R01.  Google Scholar

[11]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4$^{th}$ edition, Springer Nature Switzerland AG, 2019. doi: 10.1007/978-3-030-30351-8.  Google Scholar

[12]

D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering, IMA J. Appl. Math., 31 (1983), 253-259.  doi: 10.1093/imamat/31.3.253.  Google Scholar

[13]

H. A. Diao, H. Y. Liu and L. Wang, On generalized Holmgren's principle to the Lamé operator with applications to inverse elastic problems, Calculus of Variations and Partial Differential Equations, in press, 59 (2020), Paper No. 179, 50 pp. doi: 10.1007/s00526-020-01830-5.  Google Scholar

[14]

J. Elschner and M. Yamamoto, Uniqueness in inverse elastic scattering with finitely many incident waves, Inverse Problems, 26 (2010), 045005, 8pp. doi: 10.1088/0266-5611/26/4/045005.  Google Scholar

[15]

T. Gerlach and R. Kress, Uniqueness in inverse obstacle scattering with conductive boundary condition, Inverse Problems, 12 (1996), 619-625.  doi: 10.1088/0266-5611/12/5/006.  Google Scholar

[16]

D. Gintides, Local uniqueness for the inverse scattering problem in acoustics via the Faber-Krahn inequality, Inverse Problems, 21 (2005), 1195-1205.  doi: 10.1088/0266-5611/21/4/001.  Google Scholar

[17]

D. Gintides and L. Midrinos, Inverse scattering problem for a rigid scatterer or a cavity in elastodynamics, Zamm. J. Appl. Math. Mech., 91 (2011), 276-287.  doi: 10.1002/zamm.201000098.  Google Scholar

[18]

D. Gintides and M. Sini, Identification of obstacles using only the scatteres P-waves or the scattered S-waves, Inverse Probl. Imaging, 6 (2012), 39-55.  doi: 10.3934/ipi.2012.6.39.  Google Scholar

[19]

D. GintidesM. Sini and N. T. Thành, Detection of point-like scatterers using one type of scattered elastic waves, J. Comput. Appl. Math., 236 (2012), 2137-2145.  doi: 10.1016/j.cam.2011.09.036.  Google Scholar

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer, New York, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[21]

P. Hähner and G. C. Hsiao, Uniqueness theorems in inverse obstacle scattering of elastic waves, Inverse Problems, 9 (1993), 525-534.  doi: 10.1088/0266-5611/9/5/002.  Google Scholar

[22]

G. H. Hu, A. Kirsch and M. Sini, Some inverse problems arising from elastic scattering by rigid obstacles, Inverse Problems, 29 (2013), 015009, 21pp. doi: 10.1088/0266-5611/29/1/015009.  Google Scholar

[23]

G. H. HuJ. Z. Li and H. Y. Liu, Recovering complex elastic scatterers by a single far-field pattern, J. Differential Equations, 257 (2014), 469-489.  doi: 10.1016/j.jde.2014.04.007.  Google Scholar

[24]

M. Kar and M. Sini, On the inverse elastic scattering by interfaces using one type of scattered waves, J. Elasticity, 118 (2015), 15-38.  doi: 10.1007/s10659-014-9474-5.  Google Scholar

[25]

A. Kirsch and R. Kress, Uniqueness in inverse obstacle scattering, Inverse Problems, 9 (1993), 285-299.  doi: 10.1088/0266-5611/9/2/009.  Google Scholar

[26]

R. Kress, Uniqueness and numerical methods in inverse obstacle scattering, J. Physics: Conference Series, 73 (2007), 012003. doi: 10.1088/1742-6596/73/1/012003.  Google Scholar

[27]

V. D. Kupradze, Potential Methods in the Theory of Elasticity, Jerusalem: Israeli Program for Scientific Translations, 1965.  Google Scholar

[28]

V. D. Kupradze, Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, Amsterdam: North-Holland, 1979. Google Scholar

[29]

J. J. LaiH. Y. LiuJ. N. Xiao and Y. F. Xu, The decoupling of elastic waves from a weak formulation perspective, East Asian Journal on Applied Mathematics, 9 (2019), 241-251.  doi: 10.4208/eajam.080818.121018.  Google Scholar

[30]

P. D. Lax and R. S. Phillips, Scattering Theory, Pure and Applied Mathematics, Vol. 26 Academic Press, New York-London, 1967.  Google Scholar

[31]

H. Y. Liu and J. N. Xiao, Decoupling elastic waves and its applications, J. Differential Equations, 263 (2017), 4442-4480.  doi: 10.1016/j.jde.2017.05.022.  Google Scholar

[32]

H. Y. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), 515-524.  doi: 10.1088/0266-5611/22/2/008.  Google Scholar

[33]

X. D. Liu and B. Zhang, Direct and inverse obstacle scattering problems in a piecewise homogeneous medium, SIAM J. Appl. Math., 70 (2010), 3105-3120.  doi: 10.1137/090777578.  Google Scholar

[34]

X. D. Liu and B. Zhang, Inverse scattering by an inhomogeneous penetrable obstacle in a piecewise homogeneous medium, Acta Math. Sci., 32B (2012), 1281-1297.  doi: 10.1016/S0252-9602(12)60099-X.  Google Scholar

[35]

X. D. Liu, B. Zhang and G. H. Hu, Uniqueness in the inverse scattering problem in a piecewise homogeneous medium, Inverse Problems, 26 (2010), 015002, 14pp. doi: 10.1088/0266-5611/26/1/015002.  Google Scholar

[36]

P. A. Martin, On the scattering of elastic waves by an elastic inclusion in two dimensions, Quar. J. Mech. Appl. Math., 43 (1990), 275-291.  doi: 10.1093/qjmam/43.3.275.  Google Scholar

[37]

R. Potthast, A point source method for inverse acoustic and electromagnetic obstacle scattering problems, IMA J. Appl. Math., 61 (1998), 119-140.  doi: 10.1093/imamat/61.2.119.  Google Scholar

[38]

R. Potthast, On the convergence of a new Newton-type method in inverse scattering, Inverse Problems, 17 (2001), 1419-1434.  doi: 10.1088/0266-5611/17/5/312.  Google Scholar

[39]

F. L. Qu, J. Q. Yang and B. Zhang, Recovering an elastic obstacle containing embedded objects by the acoustic far-field measurements, Inverse Problems, 34 (2018), 015002, 8pp. doi: 10.1088/1361-6420/aa9c26.  Google Scholar

[40]

A. G. Ramm, New method for proving uniqueness theorems for obstacle inverse scattering problems, Appl. Math. Lett., 6 (1993), 19-21.  doi: 10.1016/0893-9659(93)90071-T.  Google Scholar

[41]

A. G. Ramm, Research anouncement uniqueness theorems for inverse obstacle scattering problems in Lipschitz domains, Appl. Anal., 59 (1995), 337-383.  doi: 10.1080/00036819508840411.  Google Scholar

[42]

L. Rondi, E. Sincich and M. Sini, Stable determination of a rigid scatterer in elastodynamics, arXiv: 2007.06864v1. Google Scholar

[43]

P. Stefanov and G. Uhlmann, Local uniqueness for the fixed energy fixed angle inverse problem in obstacle scattering, Proc. Amer. Math. Soc, 132 (2004), 1351-1354.  doi: 10.1090/S0002-9939-03-07363-5.  Google Scholar

[44]

J. Q. YangB. Zhang and H. W. Zhang, Uniqueness in inverse acoustic and electromagnetic scattering by penetrable obstacles with embedded objects, J. Differential Equations, 265 (2018), 6352-6383.  doi: 10.1016/j.jde.2018.07.033.  Google Scholar

[45]

D. Y. Zhang and Y. K. Guo, Uniqueness results on phaseless inverse acoustic scattering with a reference ball, Inverse Problems, 34 (2018), 085002, 12pp. doi: 10.1088/1361-6420/aac53c.  Google Scholar

Figure 1.  Possible choice of $ x^{*} $
Figure 2.  Possible choice of $ x^{*} $
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