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MiS Preprint

A compactness result for Landau state in thin-film micromagnetics

Radu Ignat and Felix Otto


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.

MSC Codes:
49S05, 82D40, 35A15, 35B25
compactness, singular perturbation, vortex, Néel wall, micromagnetics

Related publications

2011 Repository Open Access
Radu Ignat and Felix Otto

A compactness result for Landau state in thin-film micromagnetics

In: Annales de l'Institut Henri Poincaré / C, 28 (2011) 2, pp. 247-282