A compactness result for Landau state in thin-film micromagnetics
Annales de l'I.H.P. Analyse non linéaire, Volume 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:Ω 2 S 2 that are tangent at the boundary ∂Ω. We are interested in the behavior of minimizers as ϵ,η0. 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 ϵ,η } ϵ,η0 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.

DOI: 10.1016/j.anihpc.2011.01.001
Classification: 49S05,  82D40,  35A15,  35B25
Keywords: Compactness, Singular perturbation, Vortex, Néel wall, Micromagnetics, Ginzburg–Landau energy
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     title = {A compactness result for {Landau} state in thin-film micromagnetics},
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Ignat, Radu; Otto, Felix. A compactness result for Landau state in thin-film micromagnetics. Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 2, pp. 247-282. doi : 10.1016/j.anihpc.2011.01.001. http://www.numdam.org/articles/10.1016/j.anihpc.2011.01.001/

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