Lower bound for the energy of Bloch walls in micromagnetics

We study a 2D nonconvex and nonlocal variational model in micromagnetics. It consists in a free-energy functional defined over vector fields with values into the unit sphere S 2. This energy depends on two small parameters β and ε penalizing the divergence of the vector field and its vertical component, respectively. We are interested in the analysis of the asymptotic regime β ≪ ε ≪ 1 through the method of Γ−convergence. Finite energy configurations tend to become in-plane in the magnetic sample except in some small regions of length scale ε (called Bloch walls) where the magnetization varies rapidly between two directions on S 2. The limiting magnetizations are in-plane unit vector fields of vanishing divergence having an H 1 −rectifiable jump set. We prove that the Γ−limit energy concentrates on the jump set of the limiting configurations and the energetic cost of a jump is quadratic in the size of the jump. The exact charge of the jump is computed by a Γ−convergence analysis for 1D transition layers. Using the concept of entropies, we find lower bounds for the 2D model that coincide with the Γ−limit in 1D in some particular cases. Finally, we show that entropies are not appropriate in general for the 2D model in order to obtain the full Γ−limit.