Inverse problems for the anisotropic Maxwell equations. (English) Zbl 1226.35086

Kenig, Carlos E.; Salo, Mikko; Uhlmann, Gunther

Duke Math. J. 157, No. 2, 369-419 (2011).

Let \((M,g)\) be a compact, 3-dimensional Riemannian manifold with boundary \(\partial M\). The time-harmonic Maxwell equations read \(*dE=i\omega\mu H\) and \(*dH=-i\omega\varepsilon E\), where \(d\) denotes the exterior derivative, and \(*\) is the Hodge star operator. The inverse problem, at a fixed non-resonant frequency \(\omega>0\), consists of recovering the material parameters \(\varepsilon\) and \(\mu\) from the admittance map \(\Lambda:tE\mapsto tH\), where \(t\) is the tangential trace on forms. In Theorem 1.1, assuming admissability of the metric, uniqueness of the inverse problem is proved for complex-valued material parameters. In Theorem 1.2, which is essentially a corollary to Theorem 1.1, uniqueness of the inverse problem is shown for a Euclidean domain if the material parameters are positive definite \((1,1)\)-tensors, which belong to the conformal class of an admissable metric. The authors assert that these theorems are the first positive results on the inverse problem for time-harmonic Maxwell equations in anisotropic settings.
The admissability assumption on metrics was introduced by D. Dos Santos Ferreira and the authors in [Invent. Math. 178, No. 1, 119–171 (2009; Zbl 1181.35327)]. It is fundamental to a construction of complex geometric optics solutions. In the paper under review, unique complex geometric optics solutions for the Maxwell system are obtained by reducing to the Schrödinger equation with Hodge Laplacian. These solutions are then inserted into an integral identity, leading to the recovery of the electromagnetic parameters.

Reviewer: Sönke Hansen (Paderborn)

MSC:

35R30	Inverse problems for PDEs
35Q61	Maxwell equations

Keywords:

time-harmonic Maxwell equations; anisotropy; inverse problem; complex geometrical optics solutions

Citations:

Zbl 1181.35327

References:

[1]	P. Caro, Stable determination of the electromagnetic coefficients by boundary measurements , Inverse Problems 26 (2010), no. 105014. · Zbl 1205.78001 · doi:10.1088/0266-5611/26/10/105014
[2]	-, On an inverse problem in electromagnetism with local data: Stability and uniqueness , · Zbl 1219.35353 · doi:10.3934/ipi.2011.5.297
[3]	P. Caro, P. Ola, and M. Salo, Inverse boundary value problem for Maxwell equations with local data , Comm. Partial Differential Equations 34 (2009), 1425-1464. · Zbl 1185.35321 · doi:10.1080/03605300903296272
[4]	D. Colton and L. Päivärinta, The uniqueness of a solution to an inverse scattering problem for electromagnetic waves , Arch. Rational Mech. Anal. 119 (1992), 59-70. · Zbl 0756.35114 · doi:10.1007/BF00376010
[5]	M. Costabel, A coercive bilinear form for Maxwell’s equations , J. Math. Anal. Appl. 157 (1991), 527-541. · Zbl 0738.35095 · doi:10.1016/0022-247X(91)90104-8
[6]	D. Dos Santos Ferreira, C. E. Kenig, M. Salo, and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems , Invent. Math. 178 (2009), 119-171. · Zbl 1181.35327 · doi:10.1007/s00222-009-0196-4
[7]	L. Escauriaza and S. Vessella, “Optimal three cylinder inequalities for solutions to parabolic equations with Lipschitz leading coefficients” in Inverse Problems: Theory and Applications (Cortona/Pisa, 2002) , Contemp. Math. 333 , Amer. Math. Soc., Providence, 2003, 79-87. · Zbl 1056.35150
[8]	A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Full-wave invisibility of active devices at all frequencies , Comm. Math. Phys. 275 (2007), 749-789. · Zbl 1151.78006 · doi:10.1007/s00220-007-0311-6
[9]	-, Invisibility and inverse problems , Bull. Amer. Math. Soc. (N.S.) 46 (2009), 55-97. · Zbl 1159.35074 · doi:10.1090/S0273-0979-08-01232-9
[10]	L. Hörmander, Uniqueness theorems for second order elliptic differential equations , Comm. Partial Differential Equations 8 (1983), 21-64. · Zbl 0546.35023 · doi:10.1080/03605308308820262
[11]	V. Isakov, On uniqueness in the inverse conductivity problem with local data , Inverse Probl. Imaging 1 (2007), 95-105. · Zbl 1125.35113 · doi:10.3934/ipi.2007.1.95
[12]	M. Joshi and S. Mcdowall, Total determination of material parameters from electromagnetic boundary information , Pacific J. Math. 193 (2000), 107-129. · Zbl 1012.78012 · doi:10.2140/pjm.2000.193.107
[13]	C. E. Kenig, J. Sjöstrand, and G. Uhlmann, The Calderó n problem with partial data, Ann. of Math. (2) 165 (2007), 567-591. · Zbl 1127.35079 · doi:10.4007/annals.2007.165.567
[14]	Y. Kurylev, M. Lassas, and E. Somersalo, Maxwell’s equations with a polarization independent wave velocity: Direct and inverse problems , J. Math. Pures Appl. 86 (2006), 237-270. · Zbl 1134.35107 · doi:10.1016/j.matpur.2006.01.008
[15]	R. Leis, Zur Theorie elektromagnetischer Schwingungen in anisotropen inhomogenen Medien , Math. Z. 106 (1968), 213-224. · doi:10.1007/BF01110135
[16]	S. Mcdowall, Boundary determination of material parameters from electromagnetic boundary information , Inverse Problems 13 (1997), 153-163. · Zbl 0869.35113 · doi:10.1088/0266-5611/13/1/012
[17]	-, An electromagnetic inverse problem in chiral media , Trans. Amer. Math. Soc. 352 (2000), 2993-3013. JSTOR: · Zbl 1012.78011 · doi:10.1090/S0002-9947-00-02518-6
[18]	A. Nachman, Reconstructions from boundary measurements , Ann. of Math. (2) 128 (1988), 531-576. JSTOR: · Zbl 0675.35084 · doi:10.2307/1971435
[19]	A. Nachman and B. Street, Reconstruction in the Calderó n problem with partial data, Comm. Partial Differential Equations 35 (2010), 375-390. · Zbl 1186.35242 · doi:10.1080/03605300903296322
[20]	P. Ola, L. Päivärinta, and E. Somersalo, An inverse boundary value problem in electrodynamics , Duke Math. J. 70 (1993), 617-653. · Zbl 0804.35152 · doi:10.1215/S0012-7094-93-07014-7
[21]	-, “Inverse problems for time harmonic electrodynamics” in Inside Out: Inverse Problems , Math. Sci. Res. Inst. Publ. 47 , Cambridge Univ. Press, Cambridge, 2003, 169-191. · Zbl 1129.78307
[22]	P. Ola and E. Somersalo, Electromagnetic inverse problems and generalized Sommerfeld potentials , SIAM J. Appl. Math. 56 (1996), 1129-1145. JSTOR: · Zbl 0858.35138 · doi:10.1137/S0036139995283948
[23]	E. Somersalo, D. Isaacson, and M. Cheney, A linearized inverse boundary value problem for Maxwell’s equations , J. Comput. Appl. Math. 42 (1992), 123-136. · Zbl 0757.65128 · doi:10.1016/0377-0427(92)90167-V
[24]	Z. Q. Sun and G. Uhlmann, An inverse boundary value problem for Maxwell’s equations , Arch. Rational Mech. Anal. 119 (1992), 71-93. · Zbl 0757.35091 · doi:10.1007/BF00376011
[25]	J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem , Ann. of Math. (2) 125 (1987), 153-169. JSTOR: · Zbl 0625.35078 · doi:10.2307/1971291
[26]	M. E. Taylor, Partial Differential Equations, I: Basic Theory , Appl. Math. Sci. 115 , Springer, New York, 1996. · Zbl 0869.35002