# American Institute of Mathematical Sciences

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 and Imaging, 2021, 15 (3) : 539-554. doi: 10.3934/ipi.2021004
##### References:
 [1] A. Adams and J. F. Fournier, Sobolev Spaces, 2$^{nd}$ edition, Elsevier, Singapore, 2003. [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. [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. [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. [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. [6] A. Charalambopoulos, On the interior transmission problem in nondissipative, inhomogeneous, anisotropic elasticity, J. Elasticity, 67 (2002), 149-170.  doi: 10.1023/A:1023958030304. [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. [8] A. Charalambopoulos, A. Kirsch, K. A. Anagnostopoulos, D. 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. [9] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] D. Gintides, M. 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. [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. [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. [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. [23] G. H. Hu, J. 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. [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. [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. [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. [27] V. D. Kupradze, Potential Methods in the Theory of Elasticity, Jerusalem: Israeli Program for Scientific Translations, 1965. [28] V. D. Kupradze, Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, Amsterdam: North-Holland, 1979. [29] J. J. Lai, H. Y. Liu, J. 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. [30] P. D. Lax and R. S. Phillips, Scattering Theory, Pure and Applied Mathematics, Vol. 26 Academic Press, New York-London, 1967. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [42] L. Rondi, E. Sincich and M. Sini, Stable determination of a rigid scatterer in elastodynamics, arXiv: 2007.06864v1. [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. [44] J. Q. Yang, B. 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. [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.

show all references

##### References:
 [1] A. Adams and J. F. Fournier, Sobolev Spaces, 2$^{nd}$ edition, Elsevier, Singapore, 2003. [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. [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. [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. [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. [6] A. Charalambopoulos, On the interior transmission problem in nondissipative, inhomogeneous, anisotropic elasticity, J. Elasticity, 67 (2002), 149-170.  doi: 10.1023/A:1023958030304. [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. [8] A. Charalambopoulos, A. Kirsch, K. A. Anagnostopoulos, D. 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. [9] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] D. Gintides, M. 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. [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. [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. [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. [23] G. H. Hu, J. 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. [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. [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. [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. [27] V. D. Kupradze, Potential Methods in the Theory of Elasticity, Jerusalem: Israeli Program for Scientific Translations, 1965. [28] V. D. Kupradze, Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, Amsterdam: North-Holland, 1979. [29] J. J. Lai, H. Y. Liu, J. 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. [30] P. D. Lax and R. S. Phillips, Scattering Theory, Pure and Applied Mathematics, Vol. 26 Academic Press, New York-London, 1967. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [42] L. Rondi, E. Sincich and M. Sini, Stable determination of a rigid scatterer in elastodynamics, arXiv: 2007.06864v1. [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. [44] J. Q. Yang, B. 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. [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.
Possible choice of $x^{*}$
Possible choice of $x^{*}$
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