A compactness result for Landau state in thin-film micromagnetics

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 $\epsilon$ and $\eta$ and defined over vector fields $m:\Omega \subset {\mathbb{R}}^2\to S^2$ that are tangent at the boundary $\partial \Omega$. We are interested in the behavior of minimizers as $\epsilon ,\eta \to 0$. They tend to be in-plane away from a region of length scale $\epsilon$ (generically, an interior vortex ball or two boundary vortex balls) and of vanishing divergence, so that $S^1$-transition layers of length scale $\eta$ (Néel 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 Néel walls associated to the viscosity solution, so-called Landau state. Our main result concerns the compactness of vector fields $\{m_{\epsilon ,\eta }{\}}_{\epsilon ,\eta \downarrow 0}$ of energies close to the Landau state in the regime where a vortex is energetically more expensive than a Néel 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.