Siberian Mathematical Journal

, Volume 45, Issue 4, pp 730–739 | Cite as

A Stability Estimate for a Solution of the Problem of Determining the Dielectric Permittivity

  • V. G. Romanov


We consider the problem of determining the dielectric permittivity for a nonconducting and nonmagnetic medium. As information we take the traces of the tangential components of the electromagnetic field on the lateral surface of a cylindrical domain. These traces correspond to a solution to some direct problem for the Maxwell system. The impulse source of the current flux lies outside the domain in which the coefficient is sought. The main result of the article is a stability estimate for a solution to the inverse problem in question.

inverse problem Maxwell equations stability uniqueness 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Tikhonov A. N., "On uniqueness of a solution to the electroexploration problem," Dokl. Akad. Nauk SSSR, 69, No. 6, 797–800 (1949).Google Scholar
  2. 2.
    Romanov V. G., Inverse Problems of Mathematical Physics [in Russian], Nauka, Moscow (1984).Google Scholar
  3. 3.
    Romanov V. G. and Kabanikhin S. I., Inverse Problems for Geoelectrics [in Russian], Nauka, Moscow (1991).Google Scholar
  4. 4.
    Ramm A. G., Multidimensional Scattering Inverse Problems [Russian translation], Mir, Moscow (1994).Google Scholar
  5. 5.
    Ola P., Päivärinta L., and Somersalo E., "An inverse boundary value problem in electrodynamics," Duke Math. J., 70, 617–653 (1993).Google Scholar
  6. 6.
    Ola P. and Somersalo E., "Electromagnetic inverse problems and generalized Sommerfeld potential," SIAM J. Appl. Math., 560, 1129–1145 (1996).Google Scholar
  7. 7.
    Yakhno V. G., "Multidimensional inverse problems in ray formulation for hyperbolic equations," J. Inverse Ill-Posed Probl., 6, No. 4, 373–386 (1998).Google Scholar
  8. 8.
    Belishev M. I. and Glasman A. K., "A dynamical inverse problem for Maxwell's system: recovery of velocity in a regular zone," Algebra i Analiz, 12, No. 2, 131–187 (2000).Google Scholar
  9. 9.
    Belishev M. I., Isakov V. M., Pestov L. N., and Sharafutdinov V. A., "On the reconstruction of a metric from external electromagnetic measurements,"Dokl. Ross. Akad. Nauk, 372, No. 2, 298–300 (2000).Google Scholar
  10. 10.
    Romanov V. G., "Inverse problems for electrodynamics," Dokl. Akad. Nauk, 386, No. 3, 304–309 (2002).Google Scholar
  11. 11.
    Romanov V. G., "A stability estimate for a solution to one inverse problem for a system of Maxwell equations," in: Nonclassical Equations of Mathematical Physics, Inst. Mat. (Novosibirsk), Novosibirsk, 2002, pp. 196–205.Google Scholar
  12. 12.
    Romanov V. G., "A stability estimate for a solution to a two-dimensional inverse problem of electrodynamics," Sibirsk. Mat. Zh., 44, No. 4, 837–850 (2003).Google Scholar
  13. 13.
    Romanov V. G., "Stability estimation in the inverse problem of determining the speed of sound," Sibirsk. Mat. Zh., 40, No. 6, 1323–1338 (1999).Google Scholar
  14. 14.
    Romanov V. G., Investigation Methods for Inverse Problems, VSP, Utrecht (2002).Google Scholar
  15. 15.
    Ladyzhenskaya O. A., Boundary Value Problems of Mathematical Physics [in Russian], Nauka, Moscow (1973).Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • V. G. Romanov
    • 1
  1. 1.Sobolev Institute of MathematicsNovosibirsk

Personalised recommendations