On the divergence of the composition of irregular fields with BV functions

We introduce a family of (nonlinear) pairing measures that ensure the validity of the divergence rule for composite functions $\boldsymbol{B}(x,u(x))$, where $\boldsymbol{B}(\cdot,t)$ is a bounded divergence-measure vector field, and $u$ is a scalar function of bounded variation. The elements of the family depend on the choice of the pointwise representative of $u$ on its jump set. Beyond the standard properties, such as the Coarea and Gauss-Green formulas on sets of finite perimeter, this flexibility allows us to characterize the pairings that ensure the lower semicontinuity of the corresponding functionals along sequences converging in $L^1$ with controlled precise values. We show that these lower semicontinuous pairings arise as the relaxation of integral functionals defined in Sobolev spaces.