Abstract
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 language | English |
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Article number | 106819 |
Number of pages | 45 |
Journal | Advances in Mathematics |
Volume | 357 |
Early online date | 30 Sept 2019 |
DOIs | |
Publication status | Published - 1 Dec 2019 |
Keywords
- Concentration-compactness
- Domain walls
- Existence of minimizers
- Micromagnetics
- Nonlocal Allen-Cahn
- Topological degree
ASJC Scopus subject areas
- General Mathematics