*Submitted Paper*

**Inserted:** 27 apr 2015

**Last Updated:** 27 apr 2015

**Year:** 2015

**Links:**
preprint arxiv

**Abstract:**

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.

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