Energy minimisers of prescribed winding number in an S¹-valued nonlocal Allen-Cahn type model

We study a variational model for transition layers with an underlying functional that combines an Allen-Cahn type structure with an additional nonlocal interaction term. A transition layer is represented by a map from R to 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 [14], 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. We also prove non-existence in some other cases. Finally, in addition to existence, we prove a result for the structure of minimizers.