We prove an optimal regularity result in the two dimensional theory of soft ferromagnetic films. The associate Euler-Lagrange equation
is given by a microlocally degenerate variational inequality involving
fractional derivatives. A difference quotient type argument based on a dual formulation in terms of magnetostatic potentials yields a H\"older estimate for the uniquely determined gradient projection of the magnetization field.
Building on a transformation formula that we previously found in connection with Landau-Lifshitz-Gilbert equations, we present a strategy for utilizing compensated compactness methods in the context of Ginzburg-Landau approximation for harmonic maps and related problems in magnetism and superconductivity. Its applicability is illustrated by a number of new examples.
In ferromagnetic materials, the gyrotropic nature of Landau-Lifshitz-Gilbert dynamics and anisotropic effects from stray-field interaction lead in certain regimes effectively to a wave-type dynamic equation. In the case of soft thin films and small Gilbert damping, we investigate the motion of Néel walls and prove the existence of traveling wave solutions under a small constant forcing.