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Maxwell’s equations with a polarization independent wave velocity: direct and inverse problems. (English) Zbl 1134.35107

The article concerns direct and inverse boundary value problems for Maxwell’s equations on a compact Riemannian manifold \(M\) with boundary. For the particular case of a bounded smooth domain \(M\subset\mathbb{R}^3\), Maxwell’s equations read
\[ \text{curl}\,E(x,t)=-B_t(x,t),\quad\text{curl}\,H(x,t)=D_t(x,t) \]
with the constitutive relations
\[ D(x,t)= \varepsilon(x)E(x,t),\quad B(x,t)=\mu(x)H(x,t) \]
(where \(\varepsilon(x)\) and \(\mu(x)\) are positive \(3\times 3\)-matrices) and the initial-boundary conditions are
\[ E(x,0)=H(x,0)=0,\;n\times E=f\quad\text{(on } \partial M\times\mathbb{R}_+) \]
(where \(n\) is the unit normal vector to \(\partial M)\). The inverse problem is the problem of describing parameters \(\varepsilon(x)\) and \(\mu(x)\) having the same admittance map \(Z\:n \times E\mapsto n\times H\). It is shown that in the isotropic case, the boundary data given on an open part of the boundary determine both the domain and the (constants) \(\varepsilon\) and \(\mu\) uniquely. In general, the analysis is based on a geometrical formulation of the problem in terms of Riemannian geometry. Maxwell’s equations and the material parameters \(\varepsilon(x)\), \(\mu(x)\) are reformulated on terms of differential forms and Hodge-type operators \(*_0\) (the primary metric), \(*_\varepsilon\) and \(*_\mu\) (metrics related to \(\varepsilon (x),\mu(x))\). The article is adressed to the independent wave velocity case \(g_\varepsilon^{ij}=\alpha^4g^{ij}_\mu\) where \(\alpha=\alpha (x)>0\) is a proportionality factor (the scalar wave impedance). The main result concerns the global reconstruction of the shape of the 3-manifold \(M\), the underlying metric \(g_0\) on \(M\) and the scalar wave impedance \(\alpha\) (which is equivalent to the reconstruction of \(\varepsilon\) and \(\mu\)). Moreover, for the anisotropic inverse problem for bounded domains in \(\mathbb{R}^3\), the non-uniqueness is characterized by the class of symmetries of the material tensors. Very clear exposition.

MSC:

35R30 Inverse problems for PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
58J45 Hyperbolic equations on manifolds
78A25 Electromagnetic theory (general)
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[1] Anderson, M.; Katsuda, A.; Kurylev, Y.; Lassas, M.; Taylor, M., Geometric convergence, and Gel’fand’s inverse boundary problem, Invent. Math., 158, 261-321 (2004) · Zbl 1177.35245
[2] Birman, M.; Solomyak, M., The selfadjoint Maxwell operator in arbitrary domains, Algebra i Analiz, 1, 1, 96-110 (1989), (in Russian)
[3] Babich, V. M., The Hadamard ansatz, its analogues, generalisations and applications, St. Petersburg Math. J., 3, 5, 937-972 (1992) · Zbl 0791.35002
[4] M. Belishev, A.S. Blagoveščenskii, Direct method to solve a nonstationary inverse problem for the wave equation, in: Ill-Posed Problems of Math. Phys. and Anal., 1988, pp. 43-49 (in Russian); M. Belishev, A.S. Blagoveščenskii, Direct method to solve a nonstationary inverse problem for the wave equation, in: Ill-Posed Problems of Math. Phys. and Anal., 1988, pp. 43-49 (in Russian)
[5] Belishev, M., An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297, 524-527 (1987)
[6] Belishev, M.; Isakov, V.; Pestov, L.; Sharafutdinov, V., On the reconstruction of a metric from external electromagnetic measurements, Dokl. Akad. Nauk, 372, 3, 298-300 (2000) · Zbl 1048.35132
[7] Belishev, M.; Kurylev, Y., To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations, 17, 767-804 (1992) · Zbl 0812.58094
[8] Belishev, M.; Glasman, A., A dynamic inverse problem for the Maxwell system: reconstruction of the velocity in the regular zone (the BC-method), Algebra i Analiz. Algebra i Analiz, St. Petersburg Math. J., 12, 279-316 (2001), transl. in
[9] Belishev, M.; Glasman, A., Boundary control of the Maxwell dynamical system: lack of controllability by topological reasons, (Mathematical and Numerical Aspects of Wave Propagation WAVES 2003 (2003), Springer-Verlag: Springer-Verlag Berlin), 177-182 · Zbl 1060.93049
[10] Belishev, M.; Isakov, V., On the uniqueness of the reconstruction of the parameters of the Maxwell system from dynamic boundary data, Zap. Nauchn. Sem. POMI, 285, 15-32 (2002), (in Russian)
[11] Bossavit, A., Électromagnétisme, en vue de la modélisation, Mathématiques & Applications, vol. 14 (1993), Springer-Verlag: Springer-Verlag Berlin/New York, xiv+174 pp · Zbl 0787.65090
[12] Bossavit, A., Computational Electromagnetism. Variational Formulations, Complementarity, Edge Elements (1998), Academic Press Inc.: Academic Press Inc. San Diego, CA · Zbl 0945.78001
[13] Calderón, A.-P., On an inverse boundary value problem, (Seminar on Numerical Analysis and its Applications to Continuum Physics. Seminar on Numerical Analysis and its Applications to Continuum Physics, (Rio de Janeiro, 1980) (1980), Soc. Brasil. Mat.: Soc. Brasil. Mat. Rio de Janeiro), 65-73
[14] Colton, D.; Päivärinta, L., The uniqueness of a solution to an inverse scattering problem for electromagnetic waves, Arch. Rat. Mech. Anal., 119, 59-70 (1992) · Zbl 0756.35114
[15] Courant, R.; Hilbert, D., Methods of Mathematical Physics, vol. II: Partial Differential Equations (1962), Interscience: Interscience New York · Zbl 0099.29504
[16] M. Dahl, Electromagnetic Gaussian beams and Riemannian geometry, in preparation; M. Dahl, Electromagnetic Gaussian beams and Riemannian geometry, in preparation
[17] Eller, M.; Isakov, V.; Nakamura, G.; Tataru, D., Uniqueness and stability in the Cauchy problem for Maxwell’s and elasticity systems, Nonlinear PDE and Their Applications. Collège de France Seminar, XIV, 329-349 (2002) · Zbl 1038.35159
[18] Frankel, T., The Geometry of Physics (1997), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, UK, 654 pp
[19] Gel’fand, I. M., Some aspects of functional analysis and algebra, Amsterdam 1957. Amsterdam 1957, Proc. Intern. Cong. Math., 1, 253-277 (1954)
[20] Greenleaf, A.; Uhlmann, G., Local uniqueness for the Dirichlet-to-Neumann map via the two-plane transform, Duke Math. J., 108, 599-617 (2001) · Zbl 1013.35085
[21] V. Isakov, Carleman type estimates and their applications, in: K. Binghham, Y. Kurylev, E. Somersalo (Eds.), New Analytical and Geometric Methods in Inverse Problems, Springer-Verlag, Berlin/New York, pp. 93-126; V. Isakov, Carleman type estimates and their applications, in: K. Binghham, Y. Kurylev, E. Somersalo (Eds.), New Analytical and Geometric Methods in Inverse Problems, Springer-Verlag, Berlin/New York, pp. 93-126 · Zbl 1062.35165
[22] H. Isozaki, G. Uhlmann, Hyperbolic geometry and the Dirichlet-to-Neumann map, Advances in Math., submitted for publication; H. Isozaki, G. Uhlmann, Hyperbolic geometry and the Dirichlet-to-Neumann map, Advances in Math., submitted for publication · Zbl 1062.35172
[23] Kabanikhin, S.; Satybaev, A.; Shishlenin, M., Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems (2005), VSP, 179 pp · Zbl 1069.65105
[24] Kachalov, A. P., Gaussian beams for the Maxwell equations on a manifold, Zap. Nauchn. Sem. POMI, 285, 58-87 (2002), (in Russian)
[25] Kachalov, A. P., Nonstationary electromagnetic Gaussian beams in a nonhomogeneous anisotropic medium, Zap. Nauchn. Sem. POMI, 264, 83-100 (2000) · Zbl 1003.78001
[26] Kachalov, A.; Kurylev, Y., Multidimensional inverse problem with incomplete boundary spectral data, Comm. Partial Differential Equations, 23, 55-95 (1998) · Zbl 0904.65114
[27] Katchalov, A.; Kurylev, Y.; Lassas, M., Inverse Boundary Spectral Problems, Pure and Applied Mathematics, vol. 123 (2001), Chapman Hall/CRC, 290 pp · Zbl 1037.35098
[28] Katchalov, A.; Kurylev, Y.; Lassas, M., Energy measurements and equivalence of boundary data for inverse problems on non-compact manifolds, (Croke, C.; Lasiecka, I.; Uhlmann, G.; Vogelius, M., Geometric Methods in Inverse Problems and PDE Control. Geometric Methods in Inverse Problems and PDE Control, Mathematics and Applications, vol. 137 (2003), Inst. of Math. Appl.), 183-214 · Zbl 1061.35166
[29] Katsuda, A.; Kurylev, Y.; Lassas, M., Stability and reconstruction in Gel’fand inverse boundary spectral problem, (Binghham, K.; Kurylev, Y.; Somersalo, E., New Analytical and Geometric Methods in Inverse Problems (2003), Springer-Verlag: Springer-Verlag Berlin/New York), 309-320 · Zbl 1063.58017
[30] Katz, D.; Thiele, E.; Taflove, A., Validation and extension to three dimensions of the Berenger PML absorbing boundary condition for FD-TD meshes, Microwave and Guided Wave Letters, 4, 8, 268-270 (1994)
[31] Kee, C.; Kim, J.; Park, H.; Lim, H., Roles of wave impedance and refractive index in photonic crystals with magnetic and dielectric properties, IEEE Transactions on Microwave Theory and Techniques, 47, 11, 2148-2150 (1999)
[32] C. Kenig, J. Sjoestrand, G. Uhlmann, The Calderón problem with partial data, Ann. of Math., submitted for publication; Preprint arXiv math.AP/0405486; C. Kenig, J. Sjoestrand, G. Uhlmann, The Calderón problem with partial data, Ann. of Math., submitted for publication; Preprint arXiv math.AP/0405486
[33] Kurylev, Y., Admissible groups of transformations that preserve the boundary spectral data in multidimensional inverse problems, Dokl. Akad. Nauk. Dokl. Akad. Nauk, Soviet Phys. Dokl., 37, 544-545 (1993), (in Russian); transl. in
[34] Kurylev, Y., A multidimensional Gel’fand-Levitan inverse boundary problem, (Differential Equations and Mathematical Physics. Differential Equations and Mathematical Physics, Birmingham, 1994 (1995), Int. Press), 117-131 · Zbl 0929.34069
[35] Kurylev, Y., An inverse boundary problem for the Schrödinger operator with magnetic field, J. Math. Phys., 36, 6, 2761-2776 (1995) · Zbl 0845.58051
[36] Kurylev, Y., Multidimensional Gel’fand inverse problem and boundary distance map, (Soga, H., Inverse Problems Related with Geometry (1997)), 1-15
[37] Kurylev, Y.; Lassas, M., Gelf’and inverse problem for a quadratic operator pencil, J. Funct. Anal., 176, 2, 247-263 (2000) · Zbl 0968.47005
[38] Kurylev, Y.; Lassas, M., Hyperbolic inverse problem with data on a part of the boundary, (Differential Equations and Mathematical Physics. Differential Equations and Mathematical Physics, Birmingham, AL, 1999. Differential Equations and Mathematical Physics. Differential Equations and Mathematical Physics, Birmingham, AL, 1999, AMS/IP Stud. Adv. Math., vol. 16 (2000), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 259-272 · Zbl 1161.58315
[39] Kurylev, Y.; Lassas, M., Hyperbolic inverse problem and unique continuation of Cauchy data of solutions along the boundary, Proc. Roy. Soc. Edinburgh, Ser. A, 132, 4, 931-949 (2002) · Zbl 1034.35152
[40] Y. Kurylev, M. Lassas, E. Somersalo, Reconstruction of a manifold-from electromagnetic boundary measurements, in: G. Alessandrini, G. Uhlmann (Eds.), Inverse Problems: Theory and Applications, in: Contemp. Math., vol. 333, 2002, pp. 147-162; Y. Kurylev, M. Lassas, E. Somersalo, Reconstruction of a manifold-from electromagnetic boundary measurements, in: G. Alessandrini, G. Uhlmann (Eds.), Inverse Problems: Theory and Applications, in: Contemp. Math., vol. 333, 2002, pp. 147-162 · Zbl 1059.35171
[41] Y. Kurylev, M. Lassas, E. Somersalo, Focusing waves in electromagnetic inverse problems, in: H. Isozaki (Ed.), Inverse Problems and Spectral Theory, in: Contemp. Math., vol. 348, 2004, pp. 11-22; Y. Kurylev, M. Lassas, E. Somersalo, Focusing waves in electromagnetic inverse problems, in: H. Isozaki (Ed.), Inverse Problems and Spectral Theory, in: Contemp. Math., vol. 348, 2004, pp. 11-22 · Zbl 1062.35174
[42] Landau, L.; Lifshitz, E., Course of Theoretical Physics, vol. 2, The Classical Theory of Fields (1975), Pergamon Press: Pergamon Press Elmsford, NY, xiv+402 pp
[43] Langer, R. E., An inverse problem in differential equations, Bull. Amer. Math. Soc., 39, 814-820 (1933) · Zbl 0008.04603
[44] Lasiecka, I.; Triggiani, R., Control Theory for Partial Differential Equations (2000), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, UK, pp. 645-1067 · Zbl 0961.93003
[45] Lassas, M., Inverse boundary spectral problem for non-selfadjoint Maxwell’s equations with incomplete data, Comm. Partial Differential Equations, 23, 629-648 (1998) · Zbl 0906.35116
[46] Lassas, M., The impedance imaging problem as a low-frequency limit, Inverse Problems, 13, 6, 1503-1518 (1997) · Zbl 0903.35090
[47] Lassas, M.; Taylor, M.; Uhlmann, G., The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. Anal. Geom., 11, 207-222 (2003) · Zbl 1077.58012
[48] Lassas, M.; Uhlmann, G., Determining Riemannian manifold from boundary measurements, Ann. Sci. Ecole Norm. Sup. (4), 34, 5, 771-787 (2001) · Zbl 0992.35120
[49] Lee, J.; Uhlmann, G., Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42, 1097-1112 (1989) · Zbl 0702.35036
[50] Lindell, I., Methods for Electromagnetic Field Analysis (1995), Oxford Univ. Press: Oxford Univ. Press London, UK
[51] Lindell, I.; Olyslager, F., Analytic Green dyadic for a class of nonreciprocal anisotropic media, IEEE Trans. on Antennas and Propagation, 45, 10, 1563-1565 (1997)
[52] Lurie, K., The problem of effective parameters of a mixture of two isotropic dielectrics distributed in space-time and the conservation law for wave impedance in one-dimensional wave propagation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454, 1767-1779 (1998) · Zbl 0915.35102
[53] Nachman, A., Reconstructions from boundary measurements, Ann. of Math. (2), 128, 3, 531-576 (1988) · Zbl 0675.35084
[54] Nachman, A., Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2), 143, 1, 71-96 (1996) · Zbl 0857.35135
[55] Nachman, A.; Sylvester, J.; Uhlmann, G., An \(n\)-dimensional Borg-Levinson theorem, Comm. Math. Phys., 115, 4, 595-605 (1988) · Zbl 0644.35095
[56] Novikov, R.; Khenkin, G., The \(\overline{\partial} \)-equation in the multidimensional inverse scattering problem, Uspekhi Mat. Nauk, 42, 3, 93-152 (1987), (in Russian)
[57] Ola, P.; Päivärinta, L.; Somersalo, E., An inverse boundary value problem in electrodynamics, Duke Math. J., 70, 3, 617-653 (1993) · Zbl 0804.35152
[58] Ola, P.; Somersalo, E., Electromagnetic inverse problems and generalized Sommerfeld potentials, SIAM J. Appl. Math., 56, 4, 1129-1145 (1996) · Zbl 0858.35138
[59] Paquet, L., Mixed problems for the Maxwell system, Ann. Fac. Sci. Toulouse Math. (5), 4, 2, 103-141 (1982), (in French) · Zbl 0529.58038
[60] Picard, R., On the low frequency asymptotics in electromagnetic theory, J. Reine Angew. Math., 394, 50-73 (1984) · Zbl 0541.35049
[61] Romanov, V. G., An inverse problem of electrodynamics, Dokl. Matem., 66, 2, 200-205 (2002) · Zbl 1146.35421
[62] Russell, D., Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev., 20, 639-739 (1978) · Zbl 0397.93001
[63] Schlichter, L. B., An inverse boundary value problem in electrodynamics, Physics, 4, 411-418 (1953) · JFM 59.1606.02
[64] Schwarz, G., Hodge Decomposition—A Method for Solving Boundary Value Problems (1995), Springer-Verlag: Springer-Verlag Berlin/New York, 155 pp · Zbl 0828.58002
[65] Somersalo, E.; Isaacson, D.; Cheney, M., A linearized inverse boundary value problem for Maxwell’s equations, J. Comp. Appl. Math., 42, 123-136 (1992) · Zbl 0757.65128
[66] Sylvester, J., An anisotropic inverse boundary value problem, Comm. Pure Appl. Math., 43, 2, 201-232 (1990) · Zbl 0709.35102
[67] Sylvester, J., Linearizations of anisotropic inverse problems, (Päivärinta, L.; Somersalo, E., Inverse Problems in Mathematical Physics. Inverse Problems in Mathematical Physics, Lecture Notes in Phys., vol. 422 (1993), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0790.35114
[68] Sylvester, J.; Uhlmann, G., A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125, 1, 153-169 (1987) · Zbl 0625.35078
[69] Tataru, D., Unique continuation for solutions to PDEs; between Hörmander’s theorem and Holmgren’s theorem, Comm. Partial Differential Equations, 20, 5-6, 855-884 (1995) · Zbl 0846.35021
[70] Tataru, D., Unique continuation for operators with partially analytic coefficients, J. Math. Pures Appl. (9), 78, 5, 505-521 (1999) · Zbl 0936.35038
[71] Whitney, H., Geometric Integration Theory (1957), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0083.28204
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