%0 Journal Article
%T A necessary condition for lower semicontinuity of line energies
%A Pierre Bochard
%A Antonin Monteil
%J Mathematics
%D 2015
%I arXiv
%X 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.
%U http://arxiv.org/abs/1503.01021v2