Most current modeling of soft thin films is based on direct micromagnetic simulation. This is demanding due to the long-range nature of dipolar interactions, and to the necessity of resolving several small length scales simultaneously. Thus, it is natural to try and complement this effort by seeking a more analytical understanding of domain patterns, starting from the study of ground state. The motivation is not that a ferromagnet easily reaches its ground state, but rather that the most robust features of the ground state may be shared by all physically accessible local mimima, see DeSimone, Kohn, Müller, Otto and Schäfer, Low energy domain patterns in soft ferromagnetic films.

Somewhat surprisingly, the huge variety of magnetic microstructures seems to be well captured by the minimizers of a single fuctional, the energy functional of micromagnetics. Depending on the relative strength of verious material or geometric parameters, different patterns, ususally referred to as domain patterns, emerge. A common feature is that they span a broad range of well separated length-scales (the sizes of domains, walls, internal wall structure, Bloch lines, etc.). This allows for the derivation of reduced models, targeted at capturing a few length-scales at a time, and which are valid for extreme values of the material and geometric parameters.

Our work on soft ferromagnetic materials falls in this mould. By use of gamma-convergence, we have derived a reduced two-dimensional model for the response of soft films to applied magnetic films. Moreover, we have developed a numerical algorithm for the solution of the minimization problem for the reduced functional, and compared the predictions of our theory with experimental observations of Permalloy films with square cross section.