A Rigorous Derivation of a Free-Boundary Problem Arising in Superconductivity

نویسندگان

  • Etienne Sandier
  • Sylvia Serfaty
چکیده

We study the Ginzburg-Landau energy of superconductors submitted to a possibly non-uniform magnetic eld, in the limit of a large Ginzburg-Landau parameter. We prove that the induced magnetic elds associated to minimizers of the energy-functional converge as ! +1 to the solution of a free-boundary problem. This free boundary-problem has a nontrivial solution only when the applied magnetic eld is of the order of the \\rst critical eld", i.e. of the order of log. In other cases, our results are contained in those we had previously obtained ((SS2, S1, SS1]). We also derive a convergence result for the density of vortices. The Ginzburg-Landau model was introduced in the fties by Ginzburg and Landau as a phenomenological model of superconductivity. In this model, the Gibbs energy of super-conducting material, submitted to an external magnetic eld is, in a suitable normalization, 1 Here, is the domain occupied by the superconductor, is a dimensionless constant (the Ginzburg-Landau parameter) depending only on characteristic lengths of the material and of temperature. h ex is the applied magnetic eld, A : 7 ! R 3 is the vector-potential, and the induced magnetic eld in the material is h = curlA. r A = r ? iA is the associated covariant derivative. The complex-valued function u is called the \order-parameter". It is a pseudo-wave function that indicates the local state of the material. There can be essentially two phases in a superconductor : ju(x)j ' 0 is the normal phase, ju(x)j ' 1, the superconducting phase. The Ginzburg-Landau model was based on Landau's theory of phase-transitions. Since then, the model has been justiied by the microscopic theory of Bardeen-Cooper-Schrieeer (BCS theory). ju(x)j is then understood as the local density of superconducting electron pairs, called \Cooper pairs", responsible for the superconductiv-ity phenomenon. A common simpliication, that we make, is to restrict to the two-dimensional model corresponding to a innnite cylindrical domain of section R 2 (smooth and simply connected), when the applied eld is parallel to the axis of the cylinder, and all the quantities are translation-invariant. The energy-functional then reduces to

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تاریخ انتشار 1999