Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

Request Permissions   Purchase Content 


The dynamical 3-dimensional inverse problem for the Maxwell system

Author: M. N. Demchenko
Translated by: the author
Original publication: Algebra i Analiz, tom 23 (2011), nomer 6.
Journal: St. Petersburg Math. J. 23 (2012), 943-975
MSC (2010): Primary 35R30
Published electronically: September 17, 2012
MathSciNet review: 2962180
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The problem of recovering the scalar electric permittivity and magnetic permeability (respectively, $ \varepsilon $ and $ \mu $) of a medium in a bounded domain $ \Omega \subset {\mathbb{R}}^3$ by the boundary measurements on $ \partial \Omega $ is considered. As data, the value of the velocity $ c=(\varepsilon \mu )^{-1/2}$ with its normal derivative on $ \partial \Omega $ is taken, along with the response operator, which maps the tangential part $ e_\theta \mid _{\partial \Omega \times [0,2T]}$ of the electric field on the boundary to the tangential part $ h_\theta \mid _{\partial \Omega \times [0,2T]}$ of the magnetic field ($ 2T$ is the duration of measurements). With the help of the BC-method, it is established that the described data uniquely determine $ \varepsilon $ and $ \mu $ in the near-boundary layer with optical thickness $ T$, provided that the domain satisfies some geometric condition.

References [Enhancements On Off] (What's this?)

  • 1. M. I. Belishev, V. M. Isakov, L. N. Pestov, and V. A. Sharafutdinov, On the reconstruction of a metric from external electromagnetic measurements, Dokl. Akad. Nauk 372 (2000), no. 3, 298–300 (Russian). MR 1777953
  • 2. M. I. Belishev and A. K. Glasman, A dynamic inverse problem for the Maxwell system: reconstruction of the velocity in the regular zone (the BC-method), Algebra i Analiz 12 (2000), no. 2, 131–187 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 12 (2001), no. 2, 279–316. MR 1768140
  • 3. M. I. Belishev and A. S. Blagoveshchenskiĭ, Dynamical inverse problems for the wave theory, S.-Peterburg. Gos. Univ., St. Petersburg, 1999. (Russian)
  • 4. M. I. Belishev, Recent progress in the boundary control method, Inverse Problems 23 (2007), no. 5, R1–R67. MR 2353313, 10.1088/0266-5611/23/5/R01
  • 5. M. I. Belishev and V. M. Isakov, On the uniqueness of the reconstruction of the parameters of the Maxwell system from dynamic boundary data, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 285 (2002), no. Mat. Vopr. Teor. Rasprostr. Voln. 31, 15–32, 224 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N. Y.) 122 (2004), no. 5, 3459–3469. MR 1911108, 10.1023/B:JOTH.0000034024.38243.02
  • 6. Petri Ola, Lassi Päivärinta, and Erkki Somersalo, An inverse boundary value problem in electrodynamics, Duke Math. J. 70 (1993), no. 3, 617–653. MR 1224101, 10.1215/S0012-7094-93-07014-7
  • 7. Yaroslav Kurylev, Matti Lassas, and Erkki Somersalo, Maxwell’s equations with a polarization independent wave velocity: direct and inverse problems, J. Math. Pures Appl. (9) 86 (2006), no. 3, 237–270 (English, with English and French summaries). MR 2257731, 10.1016/j.matpur.2006.01.008
  • 8. M. N. Demchenko, On the partially isometric transformation of solenoidal vector fields, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 370 (2009), no. Kraevye Zadachi Matematicheskoi Fiziki i Smezhnye Voprosy Teorii Funktsii. 40, 22–43, 219 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N. Y.) 166 (2010), no. 1, 11–22. MR 2749209, 10.1007/s10958-010-9840-1
  • 9. Rolf Leis, Initial-boundary value problems in mathematical physics, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986. MR 841971
  • 10. Matthias Eller, Symmetric hyperbolic systems with boundary conditions that do not satisfy the Kreiss-Sakamoto condition, Appl. Math. (Warsaw) 35 (2008), no. 3, 323–333. MR 2453536, 10.4064/am35-3-5
  • 11. M. Eller, V. Isakov, G. Nakamura, and D. Tataru, Uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. XIV (Paris, 1997/1998) Stud. Math. Appl., vol. 31, North-Holland, Amsterdam, 2002, pp. 329–349. MR 1936000, 10.1016/S0168-2024(02)80016-9
  • 12. Hermann Sohr, The Navier-Stokes equations, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2001. An elementary functional analytic approach. MR 1928881
  • 13. M. Belishev, On a unitary transformation in the space 𝐿₂(Ω,ℝ³) associated with the Weyl decomposition, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 275 (2001), no. Mat. Vopr. Teor. Rasprostr. Voln. 30, 25–40, 310 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N. Y.) 117 (2003), no. 2, 3900–3909. MR 1854498, 10.1023/A:1024606522660
  • 14. Yu. D. Burago and V. A. Zalgaller, Vvedenie v rimanovu geometriyu, “Nauka”, Moscow, 1994 (Russian, with Russian summary). MR 1356465
  • 15. M. Sh. Birman and M. Z. Solomyak, 𝐿₂-theory of the Maxwell operator in arbitrary domains, Uspekhi Mat. Nauk 42 (1987), no. 6(258), 61–76, 247 (Russian). MR 933995

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 35R30

Retrieve articles in all journals with MSC (2010): 35R30

Additional Information

M. N. Demchenko
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia

Keywords: Inverse problem, Maxwell system, BC-method
Received by editor(s): December 20, 2010
Published electronically: September 17, 2012
Additional Notes: Supported by RFBR (grant no. 11-01-00407-a)
Dedicated: To the memory of my father, N. P. Demchenko
Article copyright: © Copyright 2012 American Mathematical Society