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), p. 247-282 / Harvested from Numdam

Résumé

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 {\mathbb{R}}^{2}\to {S}^{2}$ that are tangent at the boundary ∂Ω. 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 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 $ {\{{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.

Détail

Publié le : 2011-01-01

MR 2784071 | Zbl 1216.49041

DOI : https://doi.org/10.1016/j.anihpc.2011.01.001
Classification:  49S05,  82D40,  35A15,  35B25

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