We are interested in some energy functionals concentrated on the
discontinuity lines of divergence-free 2D vector fields valued in the circle
S1. 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)=t3,
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 tp with 0<p<1. We also show that, in this case,
the viscosity solution corresponding to a certain convex domain is not a
minimizer.Comment: 13 page