A variational problem from micromagnetics with a nonlocal term, Roger Moser (Bath)

By | October 3, 2016

We study a model for transition layers, called Nel walls, in thin films of ferromagnetic materials. The magnetisation is represented by a map from a line to the unit circle in this model, and there is an energy functional consisting of an Allen-Cahn type term and a nonlocal term penalising a fractional Sobolev norm of the first component. As the target space is a circle, we can distinguish configurations with finite energy by their winding numbers. We study the existence of minimisers with prescribed winding number. If we had only the Allen-Cahn part of the energy, then we would be able to use the well-known theory to conclude that minimisers exist only in the simplest cases. But the nonlocal term changes the situation dramatically. In some cases, we prove existence of minimisers for any reasonable winding number, whereas in other cases, we expect an intricate pattern of existence and nonexistence. This is joint work with R. Ignat (Toulouse).