A compactness result in thin-film micromagnetics and the optimality of the Néel wall
Radu Ignat, Félix Otto · Journal of the European Mathematical Society · 2008
We study a model for the magnetization in thin ferromagnetic films.It comes as a variational problem for S 1 -valued maps m (the magnetization) of two variables x :We are interested in the behavior of minimizers as ε → 0. They are expected to be S 1 -valued maps m of vanishing distributional divergence, ∇ • m = 0, so that appropriate boundary conditions enforce line discontinuities.For finite ε > 0, these line discontinuities are approximated by smooth transition layers, the so-called Néel walls.Néel walls have a line energy density of the order 1/|ln ε|.One of the main results is that the boundedness of {|ln ε|E ε (m ε )} implies the compactness of {m ε } ε↓0 , so that indeed the limits m will be S 1 -valued and weakly divergence-free.Moreover, we show the optimality of the 1-d Néel wall under 2-d perturbations as ε ↓ 0.