A compactness result for Landau state in thin-film micromagnetics
Radu Ignat and Felix Otto
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Submission date: 14. Sep. 2010
published in: Annales de l'Institut Henri Poincaré / C, 28 (2011) 2, p. 247-282
DOI number (of the published article): 10.1016/j.anihpc.2011.01.001
MSC-Numbers: 49S05, 82D40, 35A15, 35B25
Keywords and phrases: compactness, singular perturbation, vortex, Néel wall, micromagnetics
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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 and and defined over vector fields that are tangent at the boundary . We are interested in the behavior of minimizers as . They tend to be in-plane away from a region of length scale (generically, an interior vortex ball or two boundary vortex balls) and of vanishing divergence, so that transition layers of length scale (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.