Visibility, invisibility and unique recovery of inverse electromagnetic problems with conical singularities

Huaian Diao; Xiaoxu Fei; Hongyu Liu; Ke Yang

doi:10.3934/ipi.2023043

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2024, Volume 18, Issue 3: 541-570. Doi: 10.3934/ipi.2023043

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Visibility, invisibility and unique recovery of inverse electromagnetic problems with conical singularities

1.
School of Mathematics and Key Laboratory of Symbolic Computation & Knowledge, Engineering of Ministry of Education, Jilin University, Changchun, Jilin 130012, China
2.
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410000, China
3.
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong SAR, China
4.
Yantai Economic and Technological Development Zone Experimental Middle School, No.87 Jinshajiang Road, Yantai, Shandong 264000, China

^*Corresponding author: Xiaoxu Fei
^*Corresponding author: Xiaoxu Fei

Received: March 2023

Revised: September 2023

Early access: October 11, 2023

Published online: October 11, 2023

The first author is supported by National Natural Science Foundation of China (No. 12371422) and the Fundamental Research Funds for the Central Universities, JLU (No. 93Z172023Z01). The second author is supported by NSFC/RGC Joint Research Grant No. 12161160314. The third author is supported by Hong Kong RGC General Research Funds (projects 12302919, 12301420 and 11300821) and the NSFC/RGC Joint Research Fund (project N_CityU101/21).

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Abstract

In this paper, we study time-harmonic electromagnetic scattering in two scenarios, where the anomalous scatterer is either a pair of electromagnetic sources or an inhomogeneous medium, both with compact supports. We are mainly concerned with the geometrical inverse scattering problem of recovering the support of the scatterer, independent of its physical contents, by a single far-field measurement. It is assumed that the support of the scatterer (locally) possesses a conical singularity. We establish a local characterisation of the scatterer when invisibility/transparency occurs, showing that its characteristic parameters must vanish locally around the conical point. Using this characterisation, we establish several local and global uniqueness results for the aforementioned inverse scattering problems, showing that visibility must imply unique recovery. In the process, we also establish the local vanishing property of the electromagnetic transmission eigenfunctions around a conical point under the Hölder regularity or a regularity condition in terms of Herglotz approximation.

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Mathematics Subject Classification: Primary: 78A45, 35Q61, 35P25; Secondary: 78A46, 35P25, 35R30.

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Figure 1. Schematic illustration of coronal-shape scatterers. Rigorous definition is provided in Definition 2.3. The first two are the slice plottings of two coronal-shape scatterers with many conical singularities and the third one is a 3D plotting with 4 conical singularities on its body

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[1]	G. S. Alberti and Y. Capdeboscq, Elliptic regularity theory applied to time harmonic anisotropic Maxwell's equations with less than Lipschitz complex coefficients, SIAM J. Math. Anal., 46 (2014), 998-1016. doi: 10.1137/130929539.
[2]	H. Ammari, B. Li and J. Zou, Mathematical analysis of electromagnetic scattering by dielectric nanoparticles with high refractive indices, Trans. Amer. Math. Soc., 376 (2023), 39-90. doi: 10.1090/tran/8641.
[3]	H. Ammari, B. Li and J. Zou, Mathematical analysis of electromagnetic plasmonic metasurfaces, Multiscale Model. Simul., 18 (2020), 758-797. doi: 10.1137/19M1275097.
[4]	Z. Bai, H. Diao, H. Liu and Q. Meng, Stable determination of an elastic medium scatterer by a single far-field measurement and beyond, Calc. Var. Partial Differential Equations, 61 (2022), Paper No. 170, 23 pp. doi: 10.1007/s00526-022-02278-5.
[5]	E. Blåsten, Nonradiating sources and transmission eigenfunctions vanish at corners and edges, SIAM J. Math. Anal., 50 (2018), 6255-6270. doi: 10.1137/18M1182048.
[6]	E. Blåsten, X. Li, H. Liu and Y. Wang, On vanishing and localization near cusps of transmission eigenfunctions: A numerical study, Inverse Problmes, 33 (2017), 105001. doi: 10.1088/1361-6420/aa8826.
[7]	E. Blåsten and Y.-H. Lin, Radiating and non-radiating sources in elasticity, Inverse Problems, 35 (2019), 015005. doi: 10.1088/1361-6420/aae99e.
[8]	E. Blåsten and H. Liu, On corners scattering stably and stable shape determination by a single far-field pattern, Indiana Univ. Math. J., 70 (2021), 907-947. doi: 10.1512/iumj.2021.70.8411.
[9]	E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, Inverse Problems, 36 (2020), 085005, 16 pp. doi: 10.1088/1361-6420/ab958f.
[10]	E. Blåsten and H. Liu, On vanishing near corners of transmission eigenfunctions, J. Funct. Anal., 273 (2017), 3616-3632. Addendum, arXiv: 1710.08089, 2017. doi: 10.1016/j.jfa.2017.08.023.
[11]	E. L. K. Blåsten and H. Liu, Scattering by curvatures, radiationless sources, transmission eigenfunctions, and inverse scattering problems, SIAM J. Math. Anal., 53 (2021), 3801-3837. doi: 10.1137/20M1384002.
[12]	E. Blåsten, H. Liu and J. Xiao, On an electromagnetic problem in a corner and its applications, Anal. PDE, 14 (2021), 2207-2224. doi: 10.2140/apde.2021.14.2207.
[13]	E. Blåsten, L. Päivärinta and J. Sylvester, Corners always scatter, Comm. Math. Phys., 331 (2014), 725-753. doi: 10.1007/s00220-014-2030-0.
[14]	X. Cao, H. Diao and J. Li, Some recent progress on inverse scattering problems within general polyhedral geometry, Electronic Research Archive, 29 (2021), 1753-1782. doi: 10.3934/era.2020090.
[15]	X. Cao, H. Diao and H. Liu, Determining a piecewise conductive medium body by a single far-field measurement, CSIAM Trans. Appl. Math., 1 (2020), 740-765. doi: 10.4208/csiam-am.2020-0020.
[16]	X. Cao, H. Diao, H. Liu and J. Zou, On nodal and generalized singular structures of Laplacian eigenfunctions and applications to inverse scattering problems, J. Math. Pures Appl., 143 (2020), 116-161. doi: 10.1016/j.matpur.2020.09.011.
[17]	X. Cao, H. Diao, H. Liu and J. Zou, On novel geometric structures of Laplacian eigenfunctions in $\mathbb{R}^3$ and applications to inverse problems, SIAM J. Math. Anal., 53 (2021), 1263-1294. doi: 10.1137/19M1292989.
[18]	X. Cao, H. Liu, H. Diao and J. Zou, Two single-measurement uniqueness results for inverse scattering problems within polyhedral geometries, Inverse Problems and Imaging, 16 (2022), 1501-1528. doi: 10.3934/ipi.2022023.
[19]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3nd edition, Springer-Verlag, New York, 2013. doi: 10.1007/978-1-4614-4942-3.
[20]	D. Colton and R. Kress, Looking back on inverse scattering theory, SIAM Review, 60 (2018), 779-807. doi: 10.1137/17M1144763.
[21]	H. Diao, X. Cao and H. Liu, On the geometric structures of transmission eigenfunctions with a conductive boundary condition and applications, Comm. Partial Differential Equations, 46 (2021), 630-679. doi: 10.1080/03605302.2020.1857397.
[22]	H. Diao, Y. Geng, H. Liu and Q. Yu, Geometrical characterizations of non-radiating sources at polyhedral and conical corners with applications, Commun. Math. Res., (2023). doi: 10.4208/cmr.2023-0014.
[23]	H. Diao, H. Li, H. Liu and J. Tang, Spectral properties of an acoustic-elastic transmission eigenvalue problem with applications, J. Differential Equations, 371 (2023), 629-659. doi: 10.1016/j.jde.2023.07.002.
[24]	H. Diao, H. Liu and B. Sun, On a local geometric property of the generalized elastic transmission eigenfunctions and application, Inverse Problems, 37 (2021), Paper No. 105015, 36 pp. doi: 10.1088/1361-6420/ac23c2.
[25]	H. Diao, H. 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, 59 (2020), Paper No. 179, 50 pp. doi: 10.1007/s00526-020-01830-5.
[26]	H. Diao, H. Liu and L. Wang, Further results on generalized Holmgren's principle to the Lamé operator and applications, Journal of Differential Equations, 309 (2022), 841-882. doi: 10.1016/j.jde.2021.11.039.
[27]	H. Diao, H. Liu, X. Wang and K. Yang, On vanishing and localizing around corners of electromagnetic transmission resonances, Partial Differ. Equ. Appl., 2 (2021), Paper No. 78, 20 pp. doi: 10.1007/s42985-021-00131-6.
[28]	H. Diao, H. Liu, L. Zhang and J. Zou, Unique continuation from a generalized impedance edge-corner for Maxwell's system and applications to inverse problems, Inverse Problems, 37 (2021), Paper No. 035004, 32 pp. doi: 10.1088/1361-6420/abdb42.
[29]	J. Elschner and G. Hu, Acoustic scattering from corners, edges and circular cones, Arch. Ration. Mech. Anal., 228 (2018), 653-690. doi: 10.1007/s00205-017-1202-4.
[30]	T. X. D. Ha and J. Jahn, Characterizations of strictly convex sets by the uniqueness of support points, Optimization, 68 (2019), 1321-1335. doi: 10.1080/02331934.2018.1476513.
[31]	P.-Z. Kow, S. Larson, M. Salo and H. Shahgholian, Quadrature Domains for the Helmholtz Equation with Applications to Non-scattering Phenomena, Potential Anal., (2022). doi: 10.1007/s11118-022-10054-5.
[32]	Y. Liang, H. Xiang, S. Zhang and J. Zou, Preconditioners and their analyses for edge element saddle-point systems arising from time-harmonic Maxwell's equations, Numer. Algorithms, 86 (2021), 281-302. doi: 10.1007/s11075-020-00889-7.
[33]	H. Liu, On local and global structures of transmission eigenfunctions and beyond, J. Inverse Ill-Posed Probl., 30 (2022), 287-305. doi: 10.1515/jiip-2020-0099.
[34]	H. Liu, L. Rondi and J. Xiao, Mosco convergence for $H(\mathrm{curl})$ spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems, J. Eur. Math. Soc. (JEMS), 21 (2019), 2945-2993. doi: 10.4171/JEMS/895.
[35]	H. Liu and C.-H. Tsou, Stable determination by a single measurement, scattering bound and regularity of transmission eigenfunctions, Calc. Var. Partial Differential Equations, 61 (2022), Paper No. 91, 24 pp. doi: 10.1007/s00526-022-02211-w.
[36]	H. Liu and J. Xiao, On electromagnetic scattering from a penetrable corner, SIAM J. Math. Anal., 49 (2017), 5207-5241. doi: 10.1137/16M110753X.
[37]	H. Liu, M. Yamamoto and J. Zou, Reflection principle for the Maxwell equations and its application to inverse electromagnetic scattering, Inverse Problems, 23 (2007), 2357-2366. doi: 10.1088/0266-5611/23/6/005.
[38]	H. Liu and J. Zou, On uniqueness in inverse acoustic and electromagnetic obstacle scattering problems, Journal of Physics: Conference Series, 124 (2008), 012006. doi: 10.1088/1742-6596/124/1/012006.
[39]	P. Monk, Finite Element Method for Maxwell's Equations, Oxford University Press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198508885.001.0001.
[40]	J.-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic problems, vol. 144 of Applied Mathematical Sciences, Springer-Verlag New York, 2001. doi: 10.1007/978-1-4757-4393-7.
[41]	L. Päivärinta, M. Salo and E. V. Vesalainen, Strictly convex corners scatter, Rev. Mat. Iberoam., 33 (2017), 1369-1396. doi: 10.4171/RMI/975.
[42]	M. Salo and H. Shahgholian, Free boundary methods and non-scattering phenomena, Res. Math. Sci., 8 (2021), Paper No. 58, 19 pp. doi: 10.1007/s40687-021-00294-z.