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Stable configurations in superconductivity: Uniqueness, multiplicity, and vortex-nucleation. (English) Zbl 0959.35154
Essential improvements of previous results by the same author [Commun. Contemp. Math. 1, 213-254 (1999; Zbl 0944.49007)] concerning the solutions of the two-dimensional Ginzburg-Landau equation $-\nabla^2_A u=\kappa^2u(1-|u|^2),\quad -*dh=(iu,d_Au)$ which describes the energy of a superconductor, put in a uniform magnetic field $$h_{\text{ex}}$$, are presented. The most important problem is the appearance of zeros of $$u$$ (the vortex structure) which depends on the level of the parameter $$h_{\text{ex}}$$ (a function of $$\kappa$$).
The article involves many subtle results: the existence and uniqueness of Messner-type (i.e., stable and vortexless) solutions, the existence of solutions with $$n$$ vortices of degree one for higher values of $$h_{\text{ex}}$$, and the estimates of the energy of the vortex-nucleation (the energy barrier of the Messner solution).

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 82D55 Statistical mechanical studies of superconductors
##### Keywords:
vortices; Meissner solution; nucleation; energy level
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