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

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.

Ginzburg–Landau, vortices, vortex balls construction, renormalized energy
Mathematical Subject Classification 2000
Primary: 35B25, 82D55, 35Q99, 35J20
Received: 26 March 2010
Revised: 29 September 2010
Accepted: 11 November 2010
Published: 16 February 2012

Proposed: Maciej Zworski
Étienne Sandier
Université Paris-Est
 61, Avenue du Général de Gaulle
F-94010 Créteil
Sylvia Serfaty
UPMC Université Paris 06
UMR 7598 Laboratoire Jacques-Louis Lions
F-75005 Paris
Courant Institute
New York University
251 Mercer Street
New York, NY 10012
United States