Energy minimisers of prescribed winding number in an $\mathbb{S}^1$-valued nonlocal Allen-Cahn type model

Radu Ignat; Roger Moser

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arXiv · 1810.11427

Energy minimisers of prescribed winding number in an $\mathbb{S}^1$-valued nonlocal Allen-Cahn type model

Radu Ignat·Roger Moser

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Abstract

We study a variational model for transition layers in thin ferromagnetic films with an underlying functional that combines an Allen-Cahn type structure with an additional nonlocal interaction term. The model represents the magnetisation by a map from $\mathbb{R}$ to $\mathbb{S}^1$. Thus it has a topological invariant in the form of a winding number, and we study minimisers subject to a prescribed winding number. As shown in our previous paper Ignat-Moser (JDE 2017), the nonlocal term gives rise to solutions that would not be present for a functional including only the (local) Allen-Cahn terms. We complete the picture here by proving existence of minimisers in all cases where it has been conjectured. In addition, we prove non-existence in some other cases.

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BibTeX RIS

Radu Ignat, Roger Moser. 2018-10-26. Energy minimisers of prescribed winding number in an $\mathbb{S}^1$-valued nonlocal Allen-Cahn type model. https://arxiv.org/abs/1810.11427

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