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

TL;DR

This paper investigates a variational model combining local Allen-Cahn and nonlocal interactions for magnetisation in thin ferromagnetic films, establishing existence and non-existence of minimisers with prescribed winding numbers.

Contribution

It proves the existence of minimisers for all conjectured cases and identifies conditions for non-existence in a nonlocal Allen-Cahn type model.

Findings

Existence of minimisers in all conjectured cases.

Non-existence results for certain winding numbers.

Nonlocal term introduces solutions absent in purely local models.

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.