Relaxation for partially coercive integral functionals with linear growth

We prove an integral representation theorem for the ${\mathrm{L}}^1(\mathrm{\Omega };{\mathbb{R}}^m)$-relaxation of the functional $$\mathcal{F}\text{\hspace{0.17em}⁣}:u\mapsto {\int }_{\mathrm{\Omega }}f(x,u(x),\mathrm{\nabla }u(x))\mathrm{d}\mathrm{d}x,u\in {\mathrm{W}}^{1,1}(\mathrm{\Omega };{\mathbb{R}}^m),\mathrm{\Omega }\subset {\mathbb{R}}^d\text{ open,}$$ to the space $\mathrm{B}\mathrm{V}(\mathrm{\Omega };{\mathbb{R}}^m)$ under very general assumptions, requiring principally that $f$ be Carath'eodory, partially coercive, and quasiconvex in the final variable. Our result is the first of its kind which applies to integrands which are unbounded in the $u$-variable and thus allows to treat many problems from applications. Such functionals are out of reach of the classical blow-up approach introduced by Fonseca & M"uller [Arch. Ration. Mech. Anal. 123 (1993), 1–49]. Our proof relies on an intricate truncation construction (in the $x$ and $u$ arguments simultaneously) made possible by the theory of liftings as introduced in the companion paper arXiv:1708.04165, and features techniques which could be of use for other problems featuring $u$-dependent integrands.