A compactness result for Landau state in thin-film micromagnetics
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 ${\left\{{m}_{ϵ,\eta }\right\}}_{ϵ,\eta ↓0}$ 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.

DOI : https://doi.org/10.1016/j.anihpc.2011.01.001
Classification : 49S05,  82D40,  35A15,  35B25
Mots clés : Compactness, Singular perturbation, Vortex, Néel wall, Micromagnetics, Ginzburg–Landau energy
@article{AIHPC_2011__28_2_247_0,
author = {Ignat, Radu and Otto, Felix},
title = {A compactness result for Landau state in thin-film micromagnetics},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {247--282},
publisher = {Elsevier},
volume = {28},
number = {2},
year = {2011},
doi = {10.1016/j.anihpc.2011.01.001},
zbl = {1216.49041},
mrnumber = {2784071},
language = {en},
url = {www.numdam.org/item/AIHPC_2011__28_2_247_0/}
}
Ignat, Radu; Otto, Felix. A compactness result for Landau state in thin-film micromagnetics. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 2, pp. 247-282. doi : 10.1016/j.anihpc.2011.01.001. http://www.numdam.org/item/AIHPC_2011__28_2_247_0/

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