Some Regularity Results In Ferromagnetism

The magnetism of a rigid ferromagnet occupying a spatial region Ωis described by a unit vectorfield m on Ω. The total energy of m involves several terms: the anisotropy energy imposed by the lattice structure of the material, the exchange energy discouraging very rapid local changes in m , the applied energy due to external magnetic sources, and the induced magnetic field energy. Here, while incorporating all energy terms, we show that minimizers have at most isolated singularities, usually in the interior of Ω, and that there is nice asymptotic behavior at such singularities. In contrast to related harmonic map problems, the field energy is a nonlocal term, involving a solution of Maxwell's equations with coefficients depending on m