On the relaxation of polyconvex functionals with linear growth under strict convergence in $BV$

Abstract

We consider the relaxation of polyconvex functionals with linear growth with respect to the strict convergence in the space of functions of bounded variation. These functionals appears as relaxation of $F(u,\Omega):=\int_\Omega f(\nabla u)dx$, where $u:\Omega\rightarrow \mathbb R^m$, and $f$ is polyconvex. In constrast with the case of relaxation with respect to the standard $L^1$-convergence, in the case that $\Omega$ is $2$-dimensional, we prove that the sets map $A\mapsto F(u,A)$ for $A$ open, is, for every $u\in BV(\Omega;\mathbb R^m)$, $m\geq1$, the restriction of a Borel measure. This is not true in the case $\Omega\subset\mathbb R^n$, with $n\geq3$. Using the integral representation formula for a special class of functions, we also show the presence of Cartesian maps whose relaxed area functional with respect to the $L^1$-convergence is strictly larger than the area of its graph.