Local minimizers in micromagnetics and related problems

Let O R³ be a smooth bounded domain and consider the energy functional I_e(m; O) := ?_O(1/2e|Dm|² + ?(m) + 1/2|h - m|²) dx + 1/2?_R³ |h_m|² dx. Here e > 0 is a small parameter and the admissible function m lies in the Sobolev space of vector-valued functions W^1,2 (O; R³) and satisfies the pointwise constraint |m(x)| = 1 for a.e. x ? O. The induced magnetic field h_m ? L² (R³; R³) is related to m via Maxwell's equations and the function ? : S² ? R is assumed to be a sufficiently smooth, non-negative energy density with a multi-well structure. Finally h ? R³ is a constant vector. The energy functional I_e arises from the continuum model for ferromagnetic materials known as micromagnetics developed by W.F. Brown. In this paper we aim to construct local energy minimizers for this functional. Our approach is based on studying the corresponding Euler-Lagrange equation and proving a local existence result for this equation around a fixed constant solution. Our main device for doing so is a suitable version of the implicit function theorem. We then show that these solutions are local minimizers of I_e in appropriate topologies by use of certain sufficiency theorems for local minimizers. Our analysis is applicable to a much broader class of functionals than the ones introduced above and on the way to proving our main results we reflect on some related problems.