Abstract
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 \({\varepsilon}\) , penalizing the divergence of the vector field and its vertical component, respectively. Our objective is to analyze the asymptotic regime \({\beta \ll \varepsilon \ll 1}\) 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 \({\varepsilon}\) (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.
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Ignat, R., Merlet, B. Lower Bound for the Energy of Bloch Walls in Micromagnetics. Arch Rational Mech Anal 199, 369–406 (2011). https://doi.org/10.1007/s00205-010-0325-7
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DOI: https://doi.org/10.1007/s00205-010-0325-7