Analysis & PDE

  • Anal. PDE
  • Volume 4, Number 5 (2011), 757-795.

Improved lower bounds for Ginzburg–Landau energies via mass displacement

Étienne Sandier and Sylvia Serfaty

Full-text: Open access

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.

Article information

Source
Anal. PDE, Volume 4, Number 5 (2011), 757-795.

Dates
Received: 26 March 2010
Revised: 29 September 2010
Accepted: 11 November 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731183

Digital Object Identifier
doi:10.2140/apde.2011.4.757

Mathematical Reviews number (MathSciNet)
MR2901565

Zentralblatt MATH identifier
1270.35150

Subjects
Primary: 35B25: Singular perturbations 82D55: Superconductors 35Q99: None of the above, but in this section 35J20: Variational methods for second-order elliptic equations

Keywords
Ginzburg–Landau vortices vortex balls construction renormalized energy

Citation

Sandier, Étienne; Serfaty, Sylvia. Improved lower bounds for Ginzburg–Landau energies via mass displacement. Anal. PDE 4 (2011), no. 5, 757--795. doi:10.2140/apde.2011.4.757. https://projecteuclid.org/euclid.apde/1513731183


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