Homogenization of ferromagnetic materials by François Alouges Abstract In these notes we describe the homogenzation theory for ferromagnetic materials. A new phenomenon arises, in particular to take into account the geometric constraint that the magnetization is locally saturated. 1 Introduction Ferromagnetic materials present a spontaneous magnetization. They are usually the main com- ponents of permanent magnets and are of everyday use in devices such as hard disks, magnetic tapes, cellular phones, etc. Nowadays, nonhomogeneous ferromagnetic materials are the subject of a growing interest. Actually, such configurations often combine the attributes of the constituent materials, while sometimes, their properties can be strikingly different from the properties the different constituents. To predict the magnetic behaviour of these composite materials is of prime importance for applications [9]. The main objective of this paper is to perform, in the framework of De Giorgi’s notion of Γ– convergence [4] and Allaire’s notion of two-scale convergence [2] (see also the paper by Nguet- seng [10]), a mathematical homogenization study of the Gibbs-Landau free energy functional associated to a composite periodic ferromagnetic material, i.e. a ferromagnetic material in which the heterogeneities are periodically distributed inside the ferromagnetic media. 2 The Gibbs-Landau functional A ferromagnetic material, occupying the domain Ω ⊂ R 3 presents a spontaneous magnetization, which is a vectorfield M :Ω → R 3 . The magnetization is allowed to vary over Ω, and represents a continuum of magnetized pointers of as many compass distributed along the material (see Fig. 1) Figure 1. Schematization of the magnetization inside a ferromagnetic material. Here the magnetization vector is constant throughout the sample. According to Landau and Lifschitz micromagnetic theory of ferromagnetic materials (see [3], [7], [8]), the states of a rigid single-crystal ferromagnet subject to a given external magnetic field h a , are described by a vector field, the magnetization M , verifying the so-called fundamental constraint : A ferromagnetic body is always locally saturated, i.e. there exists a positive constant M S (the saturation magnetization ) such that |M (x)| = M S for a.e. x ∈ Ω. We usually write M = M S m with m:Ω → S 2 1