Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

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

 
 

 

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
DOI: https://doi.org/10.1090/S1061-0022-2012-01224-0
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 extremal electromagnetic measurement, Dokl. Akad. Nauk 372 (2000), no. 3, 298-300. (Russian) MR 1777953 (2001j:35267)
  • 2. M. I. Belishev and A. K. Glasman, Dynamical inverse problem for the Maxwell system: recovering the velocity in the regular zone (the $ BC$-method), Algebra i Analiz 12 (2000), no. 2, 131-187; English transl., St. Petersburg Math. J. 12 (2001), no. 2, 279-316. MR 1768140 (2001i:35282)
  • 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 (2008h:93001)
  • 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), 15-32; English transl., J. Math. Sci. (N. Y.) 122 (2004), no. 5, 3459-3469. MR 1911108 (2003e:78020)
  • 6. P. Ola, L. Päivärinta, and E. Somersalo, An inverse boundary value problem in electrodynamics, Duke Math. J. 70 (1993), 617-653. MR 1224101 (94i:35196)
  • 7. Y. Kurylev, M. Lassas, and E. 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. MR 2257731 (2007g:35245)
  • 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), 22-43; English transl., J. Math. Sci. (N. Y.) 166 (2010), no. 1, 11-22. MR 2749209 (2011k:35251)
  • 9. R. Leis, Initial-boundary value problems in mathematical physics, B. G. Teubner, Stuttgart, 1986. MR 0841971 (87h:35003)
  • 10. M. Eller, Symmetric hyperbolic systems with boundary conditions that do not satisfy the Kreiss-Sakamoto condition, Appl. Math. (Warsaw) 35 (2008), 323-333. MR 2453536 (2009j:35212)
  • 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 (2004c:35399)
  • 12. H. Sohr, The Navier-Stokes equations. An elementary functional analytic approach, Birkhäuser Verlag, Basel, 2001. MR 1928881 (2004b:35265)
  • 13. M. I. Belishev, On a unitary transformation in the space $ L_2(\Omega ; \mathbb{R}^3)$ associated with the Weyl decomposition, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 275 (2001), 25-40; English transl., J. Math. Sci. (N. Y.) 117 (2003), no. 2, 3900-3909. MR 1854498 (2002k:35300)
  • 14. Yu. D. Burago and V. A. Zalgaller, An introduction to Riemannian geometry, Nauka, Moscow, 1994. (Russian) MR 1356465 (96h:53002)
  • 15. M. Sh. Birman and M. Z. Solomyak, $ L_2$-theory of the Maxwell operator in arbitrary domains, Uspekhi Mat. Nauk 42 (1987), no. 6, 61-76; English transl., Russian Math. Ssurveys 42 (1987), no. 6, 75-96. MR 0933995 (89e:35127)

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
Email: demchenko@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-2012-01224-0
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

American Mathematical Society