Beyond $BV$: new pairings and Gauss-Green formulas for measure fields with divergence measure

A new notion of pairing between measure vector fields with divergence measure and scalar functions, which are not required to be weakly differentiable, is introduced. In particular, in the case of essentially bounded divergence-measure fields, the functions may not be of bounded variation. This naturally leads to the definition of $BV$-like function classes on which these pairings are well defined. Despite the lack of fine properties for such functions, our pairings surprisingly preserve many features of the recently introduced $\lambda$-pairings (Crasta, De Cicco, Malusa 2022, arXiv:1902.06052), as coarea formula, lower semicontinuity, Leibniz rules, and Gauss-Green formulas. Moreover, in a natural way new anisotropic "degenerate" perimeters are defined, possibly allowing for sets with fractal boundary.