Relaxation of signed integral functionals in BV

For integral functionals initially defined for ${u \in {\rm W}^{1,1}(\Omega; \mathbb{R}^m)}$ by $$\int_{\Omega} f(\nabla u) \, {\rm d}x$$ we establish strict continuity and relaxation results in ${{\rm BV}(\Omega; \mathbb{R}^m)}$ . The results cover the case of signed continuous integrands ${f : \mathbb{R}^{m \times d} \to \mathbb{R}}$ of linear growth at infinity. In particular, it is not excluded that the integrands are unbounded below.