Partial regularity for the Landau-Lifshitz system

The aim of this work is to consider the partial regularity for the stationary weak solutions to the Landau-Lifshitz system of Ferromagnetic spin chain from a m-dimensional manifold M into the unit sphere S-2 of R-3. The Landau-Lifshitz system is in appearance very similar to the heat flows of harmonic maps into sphere. However the monotonicity inequality, which plays an important role in getting partial regularity, does not hold in this case. This becomes a large barrier to regularity. In the present paper we get a generalized monotonicity inequality, and find the singular set of the stationary weak solutions of Landau-Lifshitz system.