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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
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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 [3] DOI: 10.1007/BF01191614 · Zbl 0834.35014 [4] DOI: 10.1007/BF01205490 · Zbl 0608.58016
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