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Qualitative Aspects of Ginzburg-Landau Equations

  • Frank Pacard
  • Tristan Rivière
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 39)

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 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)
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.

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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Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Frank Pacard
    • 1
  • Tristan Rivière
    • 2
    • 3
  1. 1.Département de MathématiquesUniversité de Paris XIICreteil CedexFrance
  2. 2.Department of MathematicsCourant Institute of Mathematical SciencesNew YorkUSA
  3. 3.CMLA, ENS-CACHANCentre National de la Recherche Scientifique 61CachanFrance

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