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}
$\begin{array}{ccc}{F}_{\mathrm{loc}}\mathrm{\left(}\mathit{u,\Omega}\mathrm{\right)}\mathrm{:}\mathrm{=}\underset{\mathrm{\left(}{\mathit{u}}_{\mathit{j}}\mathrm{\right)}}{\mathrm{inf}}\left\{\right.\underset{\mathit{j}\mathrm{\to}\mathrm{\infty}}{\mathrm{lim}\mathrm{inf}}{\mathrm{\int}}_{\mathit{\Omega}}\mathit{f}\mathrm{\left(}\mathrm{\nabla}{\mathit{u}}_{\mathit{j}}\mathrm{\right)}\mathrm{d}\mathit{x}\left|\begin{array}{c}\\ & \\ & \end{array}\right.\left\}\right.\mathit{,}& & \end{array}$
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{$r\in[1,\frac{n}{n-1})$}
$\mathit{r}\mathrm{\in}\mathrm{\left[}\mathrm{1}\mathit{,}\frac{\mathit{n}}{\mathit{n}\mathrm{-}\mathrm{1}}\mathrm{\right)}$
, we prove that
F
loc 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*}
${F}_{\mathrm{loc}}\mathrm{\left(}\mathit{u,\Omega}\mathrm{\right)}\mathrm{\ge}{\mathrm{\int}}_{\mathit{\Omega}}\mathit{f}\mathrm{\left(}\mathrm{\nabla}\mathit{u}\mathrm{\right(}\mathit{x}\mathrm{\left)}\mathrm{\right)}\mathrm{d}\mathit{x}\mathrm{+}{\mathrm{\int}}_{\mathit{\Omega}}{\mathit{f}}^{\mathrm{\infty}}\left(\right.\frac{{\mathit{D}}^{\mathit{s}}\mathit{u}}{\mathrm{\left|}{\mathit{D}}^{\mathit{s}}\mathit{u}\mathrm{\right|}}\left)\right.\mathrm{\left|}{\mathit{D}}^{\mathit{s}}\mathit{u}\mathrm{\right|}\mathit{,}$
provided
f is quasiconvex, and the recession function
f∞ (defined as
\hbox{$ f^{\infty}(\xi):= \overline{\lim}_{t\rightarrow\infty}f(t\xi )/t$}
${\mathit{f}}^{\mathrm{\infty}}\mathrm{\left(}\mathit{\xi}\mathrm{\right)}\mathrm{:}\mathrm{=}{\overline{)\mathrm{lim}}}_{\mathit{t}\mathrm{\to}\mathrm{\infty}}\mathit{f}\mathrm{\left(}\mathit{t\xi}\mathrm{\right)}\mathit{/}\mathit{t}$
) 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].