Determination of piecewise homogeneous sources for elastic and electromagnetic waves

Jian Zhai; Yue Zhao

doi:10.3934/ipi.2022065

Article Contents

2023, Volume 17, Issue 3: 614-628. Doi: 10.3934/ipi.2022065

This issue Previous Article Stable determination of an anisotropic inclusion in the Schrödinger equation from local Cauchy data Next Article Reconstruction of acoustic source with high-curvature part

Determination of piecewise homogeneous sources for elastic and electromagnetic waves

Jian Zhai^1, and
Yue Zhao^2, ,

1.
School of Mathematical Science, Fudan University, Shanghai 200433, China
2.
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

^* Corresponding author: Yue Zhao
^* Corresponding author: Yue Zhao

Received: August 2022

Revised: November 2022

Early access: December 2022

Published online: December 2022

The second author is supported in part by NSFC No.12001222.

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Abstract

This paper is concerned with inverse source problems for the time-harmonic elastic wave equations and Maxwell's equations with a single boundary measurement at a fixed frequency. We show the uniqueness and a Lipschitz-type stability estimate under the assumption that the source function is piecewise constant on a domain which is made of a union of disjoint convex polyhedral subdomains.

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Mathematics Subject Classification: Primary: 35R30; Secondary: 78A46.

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References

[1]	R. Albanese and P. B. Monk, The inverse source problem for Maxwell's equations, Inverse Problems, 22 (2006), 1023-1035. doi: 10.1088/0266-5611/22/3/018.
[2]	G. Alessandrini, M. V. de Hoop, R. Gaburro and E. Sincich, Lipschitz stability for the electrostatic inverse boundary value problem with piecewise linear conductivities, J. Math. Pures Appl., 107 (2017), 638-664. doi: 10.1016/j.matpur.2016.10.001.
[3]	G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem, Adv. Appl. Math., 35 (2005), 207-241. doi: 10.1016/j.aam.2004.12.002.
[4]	H. Ammari, G. Bao and J. L. Fleming, An inverse source problem for Maxwell's equations in magnetoencephalography, SIAM J. Appl. Math., 62 (2002), 1369-1382. doi: 10.1137/S0036139900373927.
[5]	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), 23 pp. doi: 10.1007/s00526-022-02278-5.
[6]	G. Bao, P. Li and Y. Zhao, Stability for the inverse source problems in elastic and electromagnetic waves, J. Math. Pures Appl., 134 (2020), 122-178. doi: 10.1016/j.matpur.2019.06.006.
[7]	E. Beretta, M. V. de Hoop, E. Francini, S. Vessella and J. Zhai, Uniqueness and Lipschitz stability of an inverse boundary value problem for time-harmonic elastic waves, Inverse Problems, 33 (2017), 035013. doi: 10.1088/1361-6420/aa5bef.
[8]	E. Beretta, M. V. de Hoop and L. Qiu, Lipschitz stability of an inverse boundary value problem for a Schrödinger type equation, SIAM J. Math. Anal., 45 (2013), 679-699. doi: 10.1137/120869201.
[9]	E. Beretta and E. Francini, Lipschitz stability for the electrical impedance tomography problem: The complex case, Commun. Part. Diff. Eq., 36 (2011), 1723-1749. doi: 10.1080/03605302.2011.552930.
[10]	E. Beretta, E. Francini, A. Morassi, E. Rosset and S. Vessella, Lipschitz continuous dependence of piecewise constant Lamé coefficients from boundary data: The case of non flat interfaces, Inverse Problems, 30 (2014), 125005. doi: 10.1088/0266-5611/30/12/125005.
[11]	E. Beretta, E. Francini and S. Vessella, Uniqueness and Lipschitz stability for the identification of Lamé parameters from boundary measurements, Inverse Problems and Imaging, 8 (2014), 611-644. doi: 10.3934/ipi.2014.8.611.
[12]	E. Blåsten and Y.-H. Lin, Radiating and non-radiating sources in elasticity, Inverse Problems, 35 (2019), 015005. doi: 10.1088/1361-6420/aae99e.
[13]	E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, Inverse Problems, 36 (2020), 085005. doi: 10.1088/1361-6420/ab958f.
[14]	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.
[15]	E. Blåsten, L. Päivärinta and and J. Sylvester, Corners always scatter, Commun. Math. Phys., 331 (2014), 725-753. doi: 10.1007/s00220-014-2030-0.
[16]	N. Bleistein and J. K. Cohen, Nonuniqueness in the inverse source problem in acoustics and electromagnetics, J. Math. Phys., 18 (1977), 194-201. doi: 10.1063/1.523256.
[17]	J. Chen and Z. Chen, An adaptive perfectly matched layer technique for 3-D time-harmonic electromagnetic scattering problems, Math. Comp., 77 (2008), 673-698. doi: 10.1090/S0025-5718-07-02055-8.
[18]	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.
[19]	M. Entekhabi and V. Isakov, Increasing stability in acoustic and elastic inverse source problems, SIAM J. Math. Anal., 52 (2020), 5232-5256. doi: 10.1137/19M1279885.
[20]	A. S. Fokas, Y. Kurylev and V. Marinakis, The unique determination of neuronal currents in the brain via magnetoencephalography, Inverse Problems, 20 (2004), 1067-1082. doi: 10.1088/0266-5611/20/4/005.
[21]	G. Hu and J. Li, Uniqueness to inverse source problems in an inhomogeneous medium with a single far-field pattern, SIAM J. Math. Anal., 52 (2020), 5213-5231. doi: 10.1137/20M1325289.
[22]	V. Isakov and J.-N. Wang, Uniqueness and increasing stability in electromagnetic inverse source problems, Journal of Differential Equations, 283 (2021), 110-135. doi: 10.1016/j.jde.2021.02.035.
[23]	P.-Z. Kow and J.-N. Wang, On the characterization of nonradiating sources for the elastic waves in anisotropic inhomogeneous media, SIAM J. Appl. Math., 81 (2021), 1530-1551. doi: 10.1137/20M1386293.
[24]	P. Li and X. Yuan, An adaptive finite element DtN method for the elastic wave scattering problem, Numer. Math., 150 (2022), 993-1033. doi: 10.1007/s00211-022-01273-4.
[25]	P. Li, J. Zhai and Y. Zhao, Lipschitz stability for an inverse source scattering problem at a fixed frequency, Inverse Problems, 37 (2021), 025003. doi: 10.1088/1361-6420/abd3b4.
[26]	K. Ren and Y. Zhong, Imaging point sources in heterogeneous environments, Inverse Problems, 35 (2019), 125003. doi: 10.1088/1361-6420/ab3497.
[27]	L. Rondi, E. Sincich and M. Sini, Stable determination of a rigid scatterer in elastodynamics, SIAM J. Math. Anal., 53 (2021), 2660-2689. doi: 10.1137/20M1352867.
[28]	E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, New Jersey, 1993.
[29]	J. Sylvester, Notions of support for far fields, Inverse Problems, 22 (2006), 1273-1288. doi: 10.1088/0266-5611/22/4/010.
[30]	X. Wang, Y. Guo, J. Li and H. Liu, Two gesture-computing approaches by using electromagnetic waves, Inverse Problems and Imaging, 13 (2019), 879-901. doi: 10.3934/ipi.2019040.
[31]	P. Zhang, P. Meng, W. Yin and H. Liu, A neural network method for time-dependent inverse source problem with limited-aperture data, Journal of Computational and Applied Mathematics, 421 (2023), Paper No. 114842. doi: 10.1016/j.cam.2022.114842.