A Reduced Theory for Thin-Film Micromagnetics

Micromagnetics is a nonlocal, nonconvex variational problem. Its minimizer represents the ground-state magnetization pattern of a ferromagnetic body under a specified external field. This paper identifies a physically relevant thin-film limit and shows that the limiting behavior is described by a certain "reduced" variational problem. Our main result is the Γ-convergence of suitably scaled three-dimensional micromagnetic problems to a two-dimensional reduced problem; this implies, in particular, convergence of minimizers for any value of the external field. The reduced problem is degenerate but convex; as a result, it determines some (but not all) features of the ground-state magnetization pattern in the associated thin-film limit.