# Journal of the European Mathematical Society

Full-Text PDF (499 KB) | Metadata | Table of Contents | JEMS summary

**Volume 16, Issue 7, 2014, pp. 1377–1422**

**DOI: 10.4171/JEMS/464**

Published online: 2014-08-23

A reduced model for domain walls in soft ferromagnetic films at the cross-over from symmetric to asymmetric wall types

Lukas Döring^{[1]}, Radu Ignat

^{[2]}and Felix Otto

^{[3]}(1) Max-Planck Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany

(2) Université Paul Sabatier, Toulouse, France

(3) MPI für Mathematik in den Naturwissenschaften, Leipzig, Germany

We study the Landau-Lifshitz model for the energy of multi-scale transition layers – called "domain walls'' – in soft ferromagnetic films. Domain walls separate domains of constant magnetization vectors $m^\pm_\alpha\in\mathbb{S}^2$ that differ by an angle $2\alpha$. Assuming translation invariance tangential to the wall, our main result is the rigorous derivation of a reduced model for the energy of the optimal transition layer, which in a certain parameter regime confirms the experimental, numerical and physical predictions: The minimal energy splits into a contribution from an asymmetric, divergence-free core which performs a partial rotation in $\mathbb{S}^2$ by an angle $2\theta$, and a contribution from two symmetric, logarithmically decaying tails, each of which completes the rotation from angle $\theta$ to $\alpha$ in $\mathbb{S}^1$. The angle $\theta$ is chosen such that the total energy is minimal. The contribution from the symmetric tails is known explicitly, while the contribution from the asymmetric core is analyzed in [7].

Our reduced model is the starting point for the analysis of a bifurcation phenomenon from symmetric to asymmetric domain walls. Moreover, it allows for capturing asymmetric domain walls including their extended tails (which were previously inaccessible to brute-force numerical simulation).

*Keywords: *$\Gamma$-convergence, concentration-compactness, transition layer, bifurcation, micromagnetics

Döring Lukas, Ignat Radu, Otto Felix: A reduced model for domain walls in soft ferromagnetic films at the cross-over from symmetric to asymmetric wall types. *J. Eur. Math. Soc.* 16 (2014), 1377-1422. doi: 10.4171/JEMS/464