  We introduce a family of pairings between a bounded divergence-measure vector
field and a function $u$ of bounded variation, depending on the choice of the
pointwise representative of $u$. We prove that these pairings inherit from the
standard one, introduced in [6,10], all the main properties and features (e.g.
coarea, Leibniz and Gauss--Green formulas). We also characterize the pairings
making the corresponding functionals semicontinuous with respect to the strict
convergence in $BV$. We remark that the standard pairing in general does not
share this property.

Analysis of PDEs