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


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.


Magnetic Field Neural Network Free Energy Nonlinear Dynamics Gauge Group 
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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

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