### The space $\mathrm{BV}({S}^{2},{S}^{1})$ : minimal connection and optimal lifting

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In this paper, we study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem for ${S}^{1}$-valued maps ${m}^{\text{'}}$ (the magnetization) of two variables ${x}^{\text{'}}$: ${E}_{\epsilon}\left({m}^{\text{'}}\right)=\epsilon \int {|{\nabla}^{\text{'}}\xb7{m}^{\text{'}}|}^{2}d{x}^{\text{'}}+\frac{1}{2}\int {\left||{\nabla}^{\text{'}}{|}^{-1/2}{\nabla}^{\text{'}}\xb7{m}^{\text{'}}\right|}^{2}d{x}^{\text{'}}$. We are interested in the behavior of minimizers as $\epsilon \to 0$. They are expected to be ${S}^{1}$-valued maps ${m}^{\text{'}}$ of vanishing distributional divergence ${\nabla}^{\text{'}}\xb7{m}^{\text{'}}=0$, so that appropriate boundary conditions enforce line discontinuities. For finite $\epsilon >0$, these line discontinuities are approximated by smooth transition layers, the so-called Néel walls. Néel...

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 ${m}_{\alpha}^{\pm}\in {\mathbb{S}}^{2}$ 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...

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