Asymptotic behavior of the Landau—Lifshitz model of ferromagnetism

According to the classical theory of Weiss, Landau, and Lifshitz, on a microscopic scale a ferromagnetic body is magnetically saturated (i.e., |M| =ℳ: constant) and consists of regions in which the magnetization is uniform, separated by thin transition layers. Any stationary configuration corresponds to a minimum point of an energy functional in which a small parameterε is present. The asymptotic behavior asε → 0 is studied here. It is easy to see that any sequence of minimizers contains a subsequenceMεj that converges to a fieldM. By means of a Γ-limit procedure it is shown that this fieldM is a minimizer of a new functional containing a term proportional to the area of the surfaces separating different domains of uniform magnetization. TheC1,γ-regularity of these surfaces, forγ < 1/2, is also proved under suitable assumptions for the external magnetic field.