# Qualitative Aspects of Ginzburg-Landau Equations

Chapter

## Abstract

The Ginzburg-Landau functional was introduced by V.L. Ginzburg and L.D. Landau in [27] as a model for superconductivity. If Ω is a domain of ℝ
Here the

^{ n }which is diffeomorphic to the unit ball*B*_{1}⊃ ℝ^{ n }, the functional has the following form:$$
{{G}_{K}}(u,A): = \int_{\Omega } {|\nabla u - iAu{{|}^{2}} + \frac{{{{\kappa }^{2}}}}{2}} \int_{\Omega } {{{{(1 - |u{{|}^{2}})}}^{2}}} + \int_{\Omega } {|dA - {{h}_{{ext}}}{{|}^{2}}} .
$$

(1.1)

*condensate wave function u*is defined from Ω into ℂ, and A is a 1-form defined in Ω which represents the*potential associated to the induced magnetic field h = dA*in the material. The quantity |*u*|^{2}is nothing but the density of cooper pairs of electrons that produce the superconductivity. Finally,*h*_{ext}denotes the*external magnetic field*which is applied and then appears in the problem. The parameter*κ*> 0 is usually called the*Ginzburg-Landau parameter*.## Keywords

Symmetric Solution Qualitative Aspect Limit Profile Weighted Sobolev Space Yamabe Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media New York 2000