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 that are tangent at the boundary ∂Ω. We are interested in the behavior of minimizers as . 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 -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 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 -vector fields by -vector fields away from the vortex balls.
Keywords: 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}, mrnumber = {2784071}, zbl = {1216.49041}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2011.01.001/} }
TY - JOUR AU - Ignat, Radu AU - Otto, Felix TI - A compactness result for Landau state in thin-film micromagnetics JO - Annales de l'I.H.P. Analyse non linéaire PY - 2011 SP - 247 EP - 282 VL - 28 IS - 2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2011.01.001/ DO - 10.1016/j.anihpc.2011.01.001 LA - en ID - AIHPC_2011__28_2_247_0 ER -
%0 Journal Article %A Ignat, Radu %A Otto, Felix %T A compactness result for Landau state in thin-film micromagnetics %J Annales de l'I.H.P. Analyse non linéaire %D 2011 %P 247-282 %V 28 %N 2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2011.01.001/ %R 10.1016/j.anihpc.2011.01.001 %G en %F 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, 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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