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

Abstract

AbstractA central focus of Ginzburg‐Landau theory is the understanding and characterization of vortex configurations. On a bounded domain global minimizers, and critical states in general, of the corresponding energy functional have been studied thoroughly in the limit where 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 for external fields of up to superheating field strength.In this paper, we prove that there are constants such that given natural numbers satisfying local minimizers of the Ginzburg‐Landau functional with this many vortices exist, for fields such that 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. © 2021 Wiley Periodicals LLC.