We study properties of the functional \begin{eqnarray} \mathscr{F}_{{\rm loc}}(u,\Omega):= \inf_{(u_{j})}\bigg\{ \liminf_{j\rightarrow\infty}\int_{\Omega}f(\nabla u_{j})\ud x\, \left| \!\!\begin{array}{rl} & (u_{j})\subset W_{{\rm loc}}^{1,r}\left(\Omega, \RN\right) & u_{j}\tostar u\,\,\textrm{in }\BV\left(\Omega, \RN\right) \end{array} \right. \bigg\}, \end{eqnarray} F loc ( u,Ω ) : = inf ( u j ) lim inf j → ∞ ∫ Ω f ( ∇ u j ) d x , where u ∈ BV(Ω ;RN ), and f :RN × n → R is continuous and satisfies 0 ≤ f (ξ ) ≤ L (1 + | ξ | r ). For r ∈ [1, 2), assuming f has linear growth in certain rank-one directions, we combine a result of [A. Braides and A. Coscia, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 737–756] with a new technique involving mollification to prove an upper bound for Floc . Then, for , we prove that Floc satisfies the lower bound \begin{equation*} \scF_{{\rm loc}}(u,\Omega) \geq \int_{\Omega} f(\nabla u (x))\ud x + \int_{\Omega}\finf \bigg(\frac{D^{s}u}{|D^{s}u|}\bigg)\,|D^{s}u|, \end{equation*} F loc ( u,Ω ) ≥ ∫ Ω f ( ∇ u ( x ) ) d x + ∫ Ω f ∞ D s u | D s u | | D s u | , provided f is quasiconvex, and the recession function f ∞ (defined as ) is assumed to be finite in certain rank-one directions. The proof of this result involves adapting work by [Kristensen, Calc. Var. Partial Differ. Eqs. 7 (1998) 249–261], and [Ambrosio and Dal Maso, J. Funct. Anal. 109 (1992) 76–97], and applying a non-standard blow-up technique that exploits fine properties of BV maps. It also makes use of the fact that Floc has a measure representation, which is proved in the appendix using a method of [Fonseca and Malý, Annal. Inst. Henri Poincare Anal. Non Lineaire 14 (1997) 309–338].