  We are interested in some energy functionals concentrated on the
discontinuity lines of divergence-free 2D vector fields valued in the circle
$\mathbb{S}^1$. This kind of energy has been introduced first by P. Aviles and
Y. Giga. They show in particular that, with the cubic cost function $f(t)=t^3$,
this energy is lower semicontinuous. In this paper, we construct a
counter-example which excludes the lower semicontinuity of line energies for
cost functions of the form $t^p$ with $0<p<1$. We also show that, in this case,
the viscosity solution corresponding to a certain convex domain is not a
minimizer.
