  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 (Rivi\`ere T, Serfaty S. Limiting domain wall energy for a
problem related to micromagnetics. Comm Pure Appl Math 2001; 54(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 $u$.
