A compactness result for Landau state in thin-film micromagnetics

Ignat, Radu; Otto, Felix

Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 2, pp. 247-282.

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 ε and η and defined over vector fields $m:\Omega \subset {ℝ}^2\to S^2$ that are tangent at the boundary ∂Ω. We are interested in the behavior of minimizers as $ϵ,\eta \to 0$. They tend to be in-plane away from a region of length scale ε (generically, an interior vortex ball or two boundary vortex balls) and of vanishing divergence, so that $S^1$-transition layers of length scale η (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 ${\left\{m_{ϵ,\eta }\right\}}_{ϵ,\eta \downarrow 0}$ 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.

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