#### Vol. 4, No. 5, 2011

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Improved lower bounds for Ginzburg–Landau energies via mass displacement

### Étienne Sandier and Sylvia Serfaty

Vol. 4 (2011), No. 5, 757–795
##### 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 $o\left(1\right)$ 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.

##### Keywords
Ginzburg–Landau, vortices, vortex balls construction, renormalized energy
##### Mathematical Subject Classification 2000
Primary: 35B25, 82D55, 35Q99, 35J20