2010/11/20 by Étienne Sandier, Etienne Sandier, Sylvia Serfaty +2
Computer Science · Engineering · Mathematics · Physics and Astronomy · #35Q56 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Stability and Controllability of Differential Equations #math-ph #math.AP #math.MP #msc:35Q56
paper · pdf · doi:10.48550/arxiv.1011.4616
43 pages, to appear in "Analysis & PDE"
arxiv created 2010/11/20 · openalex publication_date 2010/11/20 · arxiv updated 2010/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove some improved estimates for the Ginzburg-Landau energy (with or without 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 localisation 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. This is used crucially in a forthcoming paper relating minimizers of the Ginzburg-Landau energy with the Abrikosov lattice. It can also serve to provide lower bounds for weighted Ginzburg-Landau energies.