Calculus of Variations and Geometric Measure Theory
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P. Bochard - A. Monteil

A necessary condition for lower semicontinuity of line energies

created by monteil on 27 Apr 2015


Submitted Paper

Inserted: 27 apr 2015
Last Updated: 27 apr 2015

Year: 2015
Links: preprint arxiv


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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