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Local minimizers for the Ginzburg-Landau energy near critical magnetic field. II. (English) Zbl 0964.49005

As in Part I [Commun. Contemp. Math. 1, No. 3, 295-333 (1999; Zbl 0944.49007)] the author studies here local minimizers of the Ginzburg-Landau energy functional (depending on \( \kappa \to + \infty)\) over some domain \( \Omega \) for superconductors in a prescribed magnetic field \( h_{ex} \). Assuming that the domain \( \Omega \) has the form of a disc the author finds and describes stable solutions of the associated equations and shows how vortices appear as \( h_{ex} \) is raised from the first critical field \( H_{c_1} \). He also studies the limit \( \kappa \to + \infty \) for \( h_{ex} = H_{c_1} \) and proves that the limiting magnetic field in the superconductor satisfies the London-type equation.
In the paper some results presented in Part I are proved. The extensive appendix contains technical details concerning the proofs.
Some open problems are posed.

MSC:

49J35 Existence of solutions for minimax problems
82D55 Statistical mechanics of superconductors
49K20 Optimality conditions for problems involving partial differential equations
35J20 Variational methods for second-order elliptic equations

Citations:

Zbl 0944.49007
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References:

[1] Abrikosov A., Soviet Phys. JETP 5 pp 1174– (1957)
[2] DOI: 10.1016/S0021-7824(98)80064-0 · Zbl 0904.35023 · doi:10.1016/S0021-7824(98)80064-0
[3] DOI: 10.1007/BF01191614 · Zbl 0834.35014 · doi:10.1007/BF01191614
[4] DOI: 10.1007/BF01205490 · Zbl 0608.58016 · doi:10.1007/BF01205490
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