A necessary condition for lower semicontinuity of line energies

TL;DR

This paper investigates the lower semicontinuity of line energies for divergence-free vector fields, providing counterexamples for certain cost functions and analyzing the implications for viscosity solutions.

Contribution

It establishes a necessary condition for lower semicontinuity of line energies and shows that for cost functions with exponent less than one, lower semicontinuity fails.

Findings

Counterexamples for $t^p$ with $0<p<1$ show non-lower semicontinuity.

Viscosity solutions may not be minimizers under these conditions.

Lower semicontinuity holds for cubic cost functions $t^3$.

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.