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

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 $F(u,\mathrm{\Omega }):={\int }_{\mathrm{\Omega }}f(\mathrm{\nabla }u)dx$, where $u:\mathrm{\Omega }\to {\mathbb{R}}^m$, and $f$ is polyconvex. In constrast with the case of relaxation with respect to the standard $L^1$-convergence, in the case that $\mathrm{\Omega }$ is $2$-dimensional, we prove that the sets map $A\mapsto F(u,A)$ for $A$ open, is, for every $u\in BV(\mathrm{\Omega };{\mathbb{R}}^m)$, $m\ge 1$, the restriction of a Borel measure. This is not true in the case $\mathrm{\Omega }\subset {\mathbb{R}}^n$, with $n\ge 3$. 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.