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

TL;DR

This paper studies the relaxation of polyconvex functionals with linear growth under strict BV convergence, revealing measure-theoretic properties in 2D and differences in higher dimensions, with implications for relaxed area functionals.

Contribution

It proves that in 2D, the relaxed functional maps are measures, and it demonstrates the strict inequality of relaxed area functionals in higher dimensions.

Findings

In 2D, the relaxed functional is a Borel measure for each BV function.

In higher dimensions, the relaxed functional map is not necessarily a measure.

Existence of Cartesian maps with relaxed area larger than the graph area.

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.