Stability of the vortex in micromagnetics and related models

TL;DR

This paper provides a quantitative analysis of vortex stability in Ginzburg-Landau models within two-dimensional domains, showing that near-minimal energy configurations are close to disks with vortices, with optimal bounds.

Contribution

It extends previous characterizations by providing a quantitative stability estimate for vortex configurations in $C^{1,1}$ domains, improving the understanding of vortex stability.

Findings

Deviation from disk shape is controlled by a power of the energy

The power controlling deviation is proven to be optimal

The Lagrangian representation is key to the analysis

Abstract

We consider line-energy models of Ginzburg-Landau type in a two-dimensional simply-connected bounded domain. Configurations of vanishing energy have been characterized by Jabin, Otto and Perthame: the domain must be a disk, and the configuration a vortex. We prove a quantitative version of this statement in the class of $C^{1,1}$ domains, improving on previous results by Lorent. In particular, the deviation of the domain from a disk is controlled by a power of the energy, and that power is optimal. The main tool is a Lagrangian representation introduced by the second author, which allows to decompose the energy along characteristic curves.