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

Radu Ignat, Roger Moser

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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.

Original languageEnglish
Article number106819
Number of pages45
JournalAdvances in Mathematics
Early online date30 Sep 2019
Publication statusPublished - 1 Dec 2019


  • Concentration-compactness
  • Domain walls
  • Existence of minimizers
  • Micromagnetics
  • Nonlocal Allen-Cahn
  • Topological degree

ASJC Scopus subject areas

  • Mathematics(all)

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