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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).

35Q55 NLS equations (nonlinear Schrödinger equations)
82D55 Statistical mechanical studies of superconductors
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