Minimal energy for the traveling waves of the Landau-Lifshitz equation

We consider nontrivial finite energy traveling waves for the Landau-Lifshitz equation with easy-plane anisotropy. Our main result is the existence of a minimal energy for these traveling waves, in dimensions two, three and four. The proof relies on a priori estimates related with the theory of harmonic maps and the connection of the Landau-Lifshitz equation with the kernels appearing in the Gross-Pitaevskii equation.