Existence of an optimal asymmetric Bloch wall of prescribed angle
Let $\Omega = \mathbb{R} \times (-1, 1)$, with coordinates $(x_1, x_3)$. For $\theta \in (0, \pi/2]$ let $\mathcal{L}_\theta$ be the set of maps $m = (m_1, m_2, m_3) \in H^1_{\mathrm{loc}}(\Omega; S^2)$ with $\int_\Omega |\nabla m|^2 < \infty$ such that $$\partial_1 m_1 + \partial_3 m_3 = 0 \text{ in } \Omega, \qquad m_3 = 0 \text{ on } \partial\Omega, \qquad \int_{\Omega \cap \{\pm x_1 > 0\}} |m - (\cos\theta, \pm\sin\theta, 0)|^2 \, dx < \infty$$ for both signs, and whose boundary winding number is $+1$: under the conformal identification $z = x_1 + i x_3 \mapsto \tanh(\pi z/4)$ of $\Omega$ with the unit disk, the boundary trace $w = m_1 + i m_2$ lies in $H^{1/2}(S^1; S^1)$ and $$\sum_{k \in \mathbb{Z}} k\, |\widehat{w}_k|^2 = 1, \qquad \widehat{w}_k = \frac{1}{2\pi} \int_0^{2\pi} w(e^{it})\, e^{-ikt}\, dt.$$ These maps are stray-field-free transition layers of asymmetric Bloch-wall type in Döring, Ignat and Otto's reduced model for domain walls in soft ferromagnetic films. They showed that $\mathcal{L}_\theta$ is nonempty, that for every map in it $(m_1, m_3)$ has a vortex singularity in $\Omega$, and that without the winding condition the infimum of the Dirichlet energy is attained (2014).
Problem. Is $\inf_{m \in \mathcal{L}_\theta} \int_\Omega |\nabla m|^2 \, dx$ attained for every $\theta \in (0, \pi/2]$?
Source: Döring, Ignat and Otto (2014), Open problem 1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP, OC
- Keywords
- AIM Open Applied Problems, domain walls, micromagnetics
References
@article{doring2014,
author = {D{\"o}ring, Lukas and Ignat, Radu and Otto, Felix},
title = {A reduced model for domain walls in soft ferromagnetic films at the cross-over from symmetric to asymmetric wall types},
journal = {J. Eur. Math. Soc. (JEMS)},
volume = {16},
number = {7},
year = {2014},
pages = {1377--1422},
doi = {10.4171/JEMS/464},
eprint = {1309.2494},
archivePrefix = {arXiv},
}
@misc{aim-asymmetric-bloch-wall-attainment,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Existence of an optimal asymmetric Bloch wall of prescribed angle}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/240-asymmetric-bloch-wall-attainment.md},
note = {Entry 240 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00575, "Existence of an optimal asymmetric Bloch wall of prescribed angle", clopen.solutions.