A compactness result for Landau state in thin-film micromagnetics

  • Radu Ignat

    Laboratoire de Mathématiques, Université Paris-Sud 11, bât. 425, 91405 Orsay, France
  • Felix Otto

    Max Planck Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig, Germany


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.

Cite this article

Radu Ignat, Felix Otto, A compactness result for Landau state in thin-film micromagnetics. Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), no. 2, pp. 247–282

DOI 10.1016/J.ANIHPC.2011.01.001