  We consider line-energy models of Ginzburg-Landau type in a two-dimensional
simply-connected bounded domain. Configurations of vanishing energy have been
characterized by Jabin, Otto and Perthame: the domain must be a disk, and the
configuration a vortex. We prove a quantitative version of this statement in
the class of $C^{1,1}$ domains, improving on previous results by Lorent. In
particular, the deviation of the domain from a disk is controlled by a power of
the energy, and that power is optimal. The main tool is a Lagrangian
representation introduced by the second author, which allows to decompose the
energy along characteristic curves.
