Advertisement

Communications in Mathematical Physics

, Volume 142, Issue 1, pp 1–23 | Cite as

A boundary value problem related to the Ginzburg-Landau model

  • Anne Boutet de Monvel-Berthier
  • Vladimir Georgescu
  • Radu Purice
Article

Abstract

We analyze the Ginzburg-Landau equation for a superconductor in the case of a 2-dimensional model: a cylindrical conductor with a magnetic field parallel to the axis. This amounts to find the extrema of the free energy
$$\mathcal{A}_\kappa = 1/2\int\limits_\Omega {[|(\nabla - iA]\Phi |^2 + |B_A |^2 + \kappa /4(|\Phi |^2 - 1)^2 ]dx,}$$
where Ω is a bounded domain with smooth boundary in ℝ2,A=(A1,A2) the vector potential,B A =∂1A2−∂2A1 the magnetic field, Φ a complex field. We describe the connected components of the maximal configuration space, i.e. of the set of all (A, Φ) with components in the Sobolev spaceH1(Ω) and such that |Φ|=1 on the boundary, modulo the action of the gauge group. In the critical case κ=1 we give a complete description of the minimal configurations in each component.

Keywords

Magnetic Field Neural Network Free Energy Nonlinear Dynamics Gauge Group 
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.
    Berger, M.: Nonlinearity and functional analysis. New York: Academic Press 1977Google Scholar
  2. 2.
    Bogomol'nyi, E. B.: J. Iardernoi Fiz.24, 449 (1976)Google Scholar
  3. 3.
    Deny, J., Lions, J.-L.: Ann. Inst. FourierV, 305–370 (1953)Google Scholar
  4. 4.
    Georgescu, V.: Ann. Mat. Pura. Appl.122, 159–198 (1979)Google Scholar
  5. 5.
    Georgescu, V.: Arch. Rat. Mech. Anal.74, 143–164 (1980)Google Scholar
  6. 6.
    Ghinzburg, V. I., Landau, L. D.: J. Exp. i Teoret. Fiz.20, (12), (1950)Google Scholar
  7. 7.
    Gilbarg, D., Trudinger, N. S.: Elliptic partial differential equations. Berlin, Heidelberg, New York: Springer 1977Google Scholar
  8. 8.
    Jaffe, A., Taubes, C.: Vortices and monopoles: Structure of static gauge theories, Boston: Birkhaüser 1980Google Scholar
  9. 9.
    Lions, J.-L., Magenes, E.: Ann. Sc. Norm. Sup. Pisca16, 1–44 (1962)Google Scholar
  10. 10.
    Lions, J.-L., Magenes, E.: Non-homogeneous boundary value problems and applications, I and II. Berlin, Heidelberg, New York: Springer 1972Google Scholar
  11. 11.
    Morrey, Ch.: Multiple integrals in the calculus of variations. Berlin, Heidelberg, New York: Springer 1966Google Scholar
  12. 12.
    Rose-Innes, A. C., Rhoderich, E. N.: Introduction to superconductivity. London: Pergamon Press 1978Google Scholar
  13. 13.
    Saint-James, D., Sarma, G., Thomas, E. J.: Type II superconductivity. London: Pergamon Press 1969Google Scholar
  14. 14.
    Schrieffer, J. R.: Theory of superconductivity New York, Amsterdam: Benjamin 1964Google Scholar
  15. 15.
    Stein, E.: Singular Integrals and differentiability Properties of functions. Princeton: Princeton University Press, 1970Google Scholar
  16. 16.
    Boutet de Monvel, A., Georgescu, V., Purice, R.: Sur un problème aux limites de la théorie de Ginzburg-Landau. C. R. Acad. Sci. Paris307, série I, 55–58 (1988)Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Anne Boutet de Monvel-Berthier
    • 1
  • Vladimir Georgescu
    • 1
    • 2
  • Radu Purice
    • 2
  1. 1.Equipe de Physique Mathématique et Géométrie, C.N.R.S.Université Paris VIIParis Cedex 05France
  2. 2.Institut de Mathématique de l'Académie des Sciences de RoumanieBucarestRoumanie

Personalised recommendations