Lower Bound for the Energy of Bloch Walls in Micromagnetics

We study a two-dimensional nonconvex and nonlocal energy in micromagnetics defined over S ²-valued vector fields. This energy depends on two small parameters, β and e{varepsilon} , penalizing the divergence of the vector field and its vertical component, respectively. Our objective is to analyze the asymptotic regime b << e << 1{beta ll varepsilon ll 1} through the method of Γ-convergence. Finite energy configurations tend to become divergence-free and in-plane in the magnetic sample except in some small regions of typical width e{varepsilon} (called Bloch walls) where the magnetization connects two directions on S ². We are interested in quantifying the limit energy of the transition layers in terms of the jump size between these directions. For one-dimensional transition layers, we show by Γ-convergence analysis that the exact line density of the energy is quadratic in the jump size. We expect the same behaviour for the two-dimensional model. In order to prove that, we investigate the concept of entropies. In the prototype case of a periodic strip, we establish a quadratic lower bound for the energy with a non-optimal constant. Then we introduce and study a special class of Lipschitz entropies and obtain lower bounds coinciding with the one-dimensional Γ-limit in some particular cases. Finally, we show that entropies are not appropriate in general for proving the expected sharp lower bound.