Preprint

Inserted: 5 oct 2025
Last Updated: 5 oct 2025

Year: 2025

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. The functionals under relaxation are of the form $F(u,\Omega):=\int_\Omega f(\nabla u)dx$, 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 give a short proof of the existence of Cartesian maps whose relaxed area functional with respect to the $L^1$-convergence is strictly larger than the area of its graph.

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