| Abstract
| - 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(bla 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} where u ∈ BV( Ω;R N) , and f:R N × 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 F loc. Then, for \hbox{$r\in[1,\frac{n}{n-1})$}, we prove that F loc satisfies the lower bound \begin{equation*} \scF_{{\rm loc}}(u,\Omega) \geq \int_{\Omega} f(bla u (x))ud x + \int_{\Omega}\finf \bigg(\frac{D^{s}u}{|D^{s}u|}\bigg)\,|D^{s}u|, end{equation*} provided f is quasiconvex, and the recession function f∞ (defined as \hbox{$ f^{\infty}(\xi):= overline{\lim}_{t\rightarrow\infty}f(t\xi )/t$}) 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 F loc has a measure representation, which is proved in the appendix using a method of [Fonseca and Malý, Annal. Inst. Henri Poincaré Anal. Non Linéaire 14 (1997) 309-338].
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