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

A central focus of Ginzburg-Landau theory is the understanding and characterization of vortex configurations. On a bounded domain Omega subset of R-2, global minimizers, and critical states in general, of the corresponding energy functional have been studied thoroughly in the limit epsilon -> 0, where epsilon > 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)vertical bar Omega vertical bar/2 pi, for external fields of up to superheating field strength. In this paper, we prove that there are constants K-1,alpha > 0 such that given natural numbers satisfying 1 <= N <= h(ex)/2 pi (vertical bar Omega vertical bar-h(ex)(-1/4)), local minimizers of the Ginzburg-Landau functional with this many vortices exist, for fields such that K-1 <= h(ex) <= 1/epsilon(alpha). 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. (c) 2021 Wiley Periodicals LLC.

Automated summary

The paper focuses on understanding and characterizing vortex configurations in Ginzburg-Landau theory. It investigates the global minimizers and critical states of the energy functional in the limit as epsilon approaches 0. The authors prove the existence of local minimizers with a specific number of vortices under certain conditions, providing detailed information about their vorticity and asymptotics.