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A boundary value problem related to the Ginzburg-Landau model

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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=(A 1,A 2) the vector potential,B A =∂1 A 2−∂2 A 1 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 spaceH 1(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.

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Authors and Affiliations

  1. Equipe de Physique Mathématique et Géométrie, C.N.R.S., Université Paris VII, Tour 45-55, 5o-Etage, 2 Place Jussieu, F-75251, Paris Cedex 05, France

    Anne Boutet de Monvel-Berthier & Vladimir Georgescu

  2. Institut de Mathématique de l'Académie des Sciences de Roumanie, Bucarest, Roumanie

    Vladimir Georgescu & Radu Purice

Authors
  1. Anne Boutet de Monvel-Berthier
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  2. Vladimir Georgescu
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  3. Radu Purice
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Communicated by H. Araki

This paper is dedicated to the memory of Michel Sirrugue

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de Monvel-Berthier, A.B., Georgescu, V. & Purice, R. A boundary value problem related to the Ginzburg-Landau model. Commun.Math. Phys. 142, 1–23 (1991). https://doi.org/10.1007/BF02099170

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  • DOI: https://doi.org/10.1007/BF02099170

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