Advertisement

Journal of Mathematical Sciences

, Volume 132, Issue 4, pp 404–418 | Cite as

Some Properties of Solutions to a Second Order Elliptic Equation with Principal Part of Divergence Form with Potential Concentrated on a Hypersurface

  • S. V. Morozov
Article

Abstract

A number of classical results of the theory of second order elliptic equations of divergence form in ℝ d, d ⩾ 2, is generalized to problems with singular potential concentrated on a hypersurface of codimension 1. Such results are required, for example, for describing photon crystals. Bibliography: 8 titles.

Keywords

Photon Crystal Elliptic Equation Divergence Form Classical Result Principal Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations [in Russian], Moscow, Nauka, 1973; English transl. of the 1st ed.: Linear and Quasilinear Elliptic Equations, New York, Academic Press, 1968.Google Scholar
  2. 2.
    D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Berlin, Springer, 1977.Google Scholar
  3. 3.
    M. Sh. Birman, R. G. Shterenberg, and T. A. Suslina, “Absolute continuity of the spectrum of a two-dimensional Schrodinger operator with potential supported on a periodic system of curves,” Algebra Anal. 12 (2000), no. 6, 140–177; English transl.: St. Petersb. Math. J. 12 (2003), no. 6, 983–1012.MathSciNetGoogle Scholar
  4. 4.
    T. A. Suslina and R. G. Shterenberg, “Absolute continuity of the spectrum of the Schrodinger operator with the potential concentrated on a periodic system of hypersurfaces,” Algebra Anal 13 (2001), no. 5, 197–240; English transl.: St. Petersb. Math. J. 13 (2002), no. 5, 859–891.Google Scholar
  5. 5.
    M. Sh. Birman, and T. A. Suslina, “Second order periodic differential operators. Threshold properties and homogenization,” Algebra Anal. 15 (2003), no. 5, 1–108; English transl.: St. Petersb. Math. J. 15 (2004), no. 5, 639–714.MathSciNetGoogle Scholar
  6. 6.
    M. Sh. Birman, Private communication.Google Scholar
  7. 7.
    T. A. Suslina, “On the homogenization of the Maxwell system” [in Russian], Algebra Anal. 16 (2004), no. 5, 162–244.MathSciNetGoogle Scholar
  8. 8.
    V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators [in Russian], Moscow, Nauka, 1993; English transl.: Homogenization of Differential Operators and Integral Functionals, Berlin, Springer-Verlag, 1994.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • S. V. Morozov

There are no affiliations available

Personalised recommendations