Local Minimizers with Unbounded Vorticity for the 2D Ginzburg‐Landau Functional

Abstract

A central focus of Ginzburg-Landau theory is the understanding and characterization of vortex configurations. On a bounded domain R 2 ; global minimizers, and critical states in general, of the corresponding energy functional have been studied thoroughly in the limit 3 0; where > 0 is the inverse of the Ginzburg-Landau parameter. A notable open problem is whether there are solutions of the Ginzburg-Landau equation for any number of vortices below h ex jj=2; for external fields of up to superheating field strength.In this paper, we prove that there are constants K 1 ; > 0 such that given natural numbers satisfyinglocal minimizers of the Ginzburg-Landau functional with this many vortices exist, for fields such that K 1 h ex 1= : Our strategy consists of combining:the minimization over a subset of configurations for which we can obtain a very precise localization of vortices; expansion of the energy in terms of a modified vortex interaction energy that allows for a reduction to a potential theory problem; and a quantitative vortex separation result for admissible configurations.Our results provide detailed information about the vorticity and refined asymptotics of the local minimizers that we construct.