

Preprint 52/2010
A compactness result for Landau state in thin-film micromagnetics
Radu Ignat and Felix Otto
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Submission date: 14. Sep. 2010
Pages: 42
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
Bibtex
MSC-Numbers: 49S05, 82D40, 35A15, 35B25
Keywords and phrases: compactness, singular perturbation, vortex, Néel wall, micromagnetics
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Abstract:
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.