The existence of Ginzburg-Landau solutions on the plane by a direct variational method

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Abstract

This paper studies the existence and the minimization problem of the solutions of the Ginzburg-Landau equations in R2 coupled with an external magnetic field or a source current. The lack of a suitable Sobolev inequality makes it necessary to consider a variational problem over a special admissible space so that the space norms of the gauge vector fields of a minimization sequence can be controlled by the corresponding energy upper bound and a solution may be obtained as a minimizer of a modified energy of the problem. Asymptotic properties and flux quantization are established for finite-energy solutions. Besides, it is shown that the solutions obtained also minimize the original Ginzburg-Landau energy when the admissible space is properly chosen.

Original languageEnglish (US)
Pages (from-to)517-536
Number of pages20
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume11
Issue number5
DOIs
StatePublished - Sep 1 2016

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Ginzburg-Landau
Direct Method
Variational Methods
Energy
Sobolev Inequality
Ginzburg-Landau Equation
Gauge Field
Minimizer
Variational Problem
Minimization Problem
External Field
Asymptotic Properties
Gages
Quantization
Vector Field
Magnetic Field
Magnetic fields
Fluxes
Upper bound
Minimise

Keywords

  • Minimization
  • Sobolev inequalities

ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics

Cite this

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abstract = "This paper studies the existence and the minimization problem of the solutions of the Ginzburg-Landau equations in R2 coupled with an external magnetic field or a source current. The lack of a suitable Sobolev inequality makes it necessary to consider a variational problem over a special admissible space so that the space norms of the gauge vector fields of a minimization sequence can be controlled by the corresponding energy upper bound and a solution may be obtained as a minimizer of a modified energy of the problem. Asymptotic properties and flux quantization are established for finite-energy solutions. Besides, it is shown that the solutions obtained also minimize the original Ginzburg-Landau energy when the admissible space is properly chosen.",
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AB - This paper studies the existence and the minimization problem of the solutions of the Ginzburg-Landau equations in R2 coupled with an external magnetic field or a source current. The lack of a suitable Sobolev inequality makes it necessary to consider a variational problem over a special admissible space so that the space norms of the gauge vector fields of a minimization sequence can be controlled by the corresponding energy upper bound and a solution may be obtained as a minimizer of a modified energy of the problem. Asymptotic properties and flux quantization are established for finite-energy solutions. Besides, it is shown that the solutions obtained also minimize the original Ginzburg-Landau energy when the admissible space is properly chosen.

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