Improved lower bounds for Ginzburg-Landau energies via mass displacement

Étienne Sandier, Sylvia Serfaty

Research output: Contribution to journalArticle

Abstract

We prove some improved estimates for the Ginzburg-Landau energy (with or without a magnetic field) in two dimensions, relating the asymptotic energy of an arbitrary configuration to its vortices and their degrees, with possibly unbounded numbers of vortices. The method is based on a localization of the "ball construction method" combined with a mass displacement idea which allows to compensate for negative errors in the ball construction estimates by energy "displaced" from close by. Under good conditions, our main estimate allows to get a lower bound on the energy which includes a finite order "renormalized energy" of vortex interaction, up to the best possible precision, i.e., with only a o(1) error per vortex, and is complemented by local compactness results on the vortices. Besides being used crucially in a forthcoming paper, our result can serve to provide lower bounds for weighted Ginzburg-Landau energies.

Original languageEnglish (US)
Pages (from-to)757-795
Number of pages39
JournalAnalysis and PDE
Volume4
Issue number5
DOIs
StatePublished - Dec 1 2011

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Keywords

  • Ginzburg-Landau
  • Renormalized energy
  • Vortex balls construction
  • Vortices

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Applied Mathematics

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