A compactness result for Landau state in thin-film micromagnetics

TL;DR

This paper analyzes the behavior of minimizers in a thin-film micromagnetics model, establishing energy bounds and compactness results as parameters tend to zero, with a focus on vortex and Nél walls configurations.

Contribution

It provides a new compactness result for Landau states in thin-film micromagnetics, using Ginzburg-Landau techniques and energy concentration analysis.

Findings

Upper bound for minimal energy including vortex and Nél walls

Compactness of minimizers near Landau states when vortices are costly

Energy concentration on vortex balls and approximation of vector fields

Abstract

We deal with a nonconvex and nonlocal variational problem coming from thin-film micromagnetics. It consists in a free-energy functional depending on two small parameters $\eps$ and $\eta$ and defined over $S^2-$vector fields $m$ that are tangent at the boundary of a two-dimensional domain $\Omega$. We are interested in the behavior of minimizers as $\eps, \eta \to 0$. The minimizers tend to be in-plane away from a region of length scale $\eps$ (generically, an interior vortex ball or two boundary vortex balls) and of vanishing divergence, so that $S^1-$transition layers of length scale $\eta$ (N\'eel walls) are enforced by the boundary condition. We first prove an upper bound for the minimal energy that corresponds to the cost of a vortex and the configuration of N\'eel walls associated to the viscosity solution, so-called Landau state. Our main result concerns the compactness of vector fields $m_{\eps, \eta}$ of energies close to the Landau state in the regime where a vortex is energetically more expensive than a N\'eel wall. Our method uses techniques developed for the Ginzburg-Landau type problems for the concentration of energy on vortex balls, together with an approximation argument of $S^2-$vector fields by $S^1-$vector fields away from the vortex balls.