2017 # Néel walls with prescribed winding number and how a nonlocal term can change the energy landscape

**Abstract:** We study a nonlocal Allen-Cahn type problem for vector fields of unit length, arising from a model for domain walls (called Néel walls) in ferromagnetism. We show that the nonlocal term gives rise to new features in the energy landscape; in particular, we prove existence of energy minimisers with prescribed winding number that would be prohibited in a local model.

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“…Some of these observations are obviously consistent with our results [19,20] about the existence/nonexistence of minimisers of the functional E 1 for a prescribed winding number. In particular, for winding numbers giving rise to the situation of statement 4, we find that E 1 has no minimiser.…”

confidence: 90%

“…Some of these observations are obviously consistent with our results [19,20] about the existence/nonexistence of minimisers of the functional E 1 for a prescribed winding number. In particular, for winding numbers giving rise to the situation of statement 4, we find that E 1 has no minimiser.…”

confidence: 90%

“…This is because otherwise, the attraction of next-to-neighbouring walls may dominate the (comparatively small) repulsion of neighbouring walls. This is a phenomenon that was exploited in other papers of the authors [19,20] (in a somewhat different model with fixed ) to construct energy minimisers comprising several Néel walls. For the problem studied here, a plausible consequence (not proved here) is that for small transition angles, several Néel walls may in the limit 0 collapse to a single composite Néel wall, corresponding to a jump of more than 2π in the lifting (i.e., if α ∈ (0, π) is small, under the condition (8), we may have a jump of size |ω n | ≥ 2π).…”

confidence: 94%