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A rigorous derivation of a free-boundary problem arising in superconductivity. (English) Zbl 1174.35552
Summary: We study the Ginzburg-Landau energy of superconductors submitted to a possibly non-uniform magnetic field, in the limit of a large Ginzburg-Landau parameter \(\kappa\). We prove that the induced magnetic fields associated to minimizers of the energy-functional converge as \(\kappa\to +\infty\) to the solution of a free-boundary problem. This free-boundary problem has a nontrivial solution only when the applied magnetic field is of the order of the “first critical field”, i.e. of the order of \(\log \kappa\). In other cases, our results are contained in those we had previously obtained [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 17, No. 1, 119–145 (2000; Zbl 0947.49004), Rev. Math. Phys. 12, No. 9, 1219–1257 (2000; Zbl 0964.49006), and Commun. Contemp. Math. 1, No. 2, 213–254 (1999; Zbl 0944.49007)]. We also derive a convergence result for the density of vortices.

MSC:
35R35 Free boundary problems for PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
82D55 Statistical mechanical studies of superconductors
35A15 Variational methods applied to PDEs
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