Qualitative Aspects of Ginzburg-Landau Equations

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 ℝⁿ which is diffeomorphic to the unit ball B ₁ ⊃ ℝⁿ, 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)

Here the 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.