Stabilization of a degenerate minimization problem with the simple-layer potential

We consider the reduced model for thin-film devices in stationary micromagnetics proposed in [4, DeSimone, Kohn, Müller, Otto, Schäfer 2001]. In the case of soft material, one of the energy contributions is negligible, and the problem becomes degenerate. The analysis and the numerical scheme recently developed in [7, Ferraz-Leite, Melenk, Praetorius 2011] are not satisfactory in this case. In the present work, we overcome the degeneracy and extend the numerical scheme by introducing a stabilizing energy term. Convergence of the method is established and a numerical experiment concludes the paper. 1 Model problem and introduction We consider the model that was proposed in [3, 4] for the simulation of thin ferromagnetic films: Let Ω = ω × (0, t) be a thin ferromagnetic sample. The surface ω ⊆ R2 is a Lipschitz domain with diam(ω) ∼ 1. We model the sample as this screen and neglect the thickness t ≪ 1. We are interested in the effective behavior of the ferromagnetic material when exposed to an in-plane exterior field hext. For simplicity we assume hext ∈ R2 to be constant. The magnetization m : ω → R2 is an in-plane vector field. In the full 3dimensional model due to Landau and Lifschits [8], the magnetization is of constant length |m(x)| = 1. In the reduced thin-film model, however, this constraint relaxes to the convex admissibility condition |m(x)| ≤ 1 almost everywhere in ω . Let V denote the simple-layer operator on ω corresponding to the Laplacian in 3D, i.e. (Vφ)(x) := 1 4π ∫ ω φ(y) |x− y| dy, x ∈ ω . (1) Samuel Ferraz-Leite Max-Planck-Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany, e-mail: [email protected]