## Analysis & PDE

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

### Improved lower bounds for Ginzburg–Landau energies via mass displacement

#### 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
Revised: 29 September 2010
Accepted: 11 November 2010
First available in Project Euclid: 20 December 2017

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

#### 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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