Rectifiability of entropy defect measures in a micromagnetics model

We study the fine properties of a class of weak solutions u of the eikonal equation arising as asymptotic domain of a family of energy functionals introduced in [T. Rivière and S. Serfaty, Limiting domain wall energy for a problem related to micromagnetics, Comm. Pure Appl. Math. 54 2001, 3, 294–338]. In particular, we prove that the entropy defect measure associated to u is concentrated on a 1-rectifiable set, which detects the jump-type discontinuities of u.