# Pairings between bounded divergence-measure vector fields and BV functions

## TL;DR

This paper introduces a new family of pairings between divergence-measure vector fields and BV functions, extending existing concepts and analyzing their properties, including semicontinuity under strict convergence.

## Contribution

It develops a generalized framework for pairings that depend on the pointwise representative of BV functions, preserving key properties and characterizing semicontinuous cases.

## Key findings

- New family of pairings introduced
- Standard pairing does not always have semicontinuity
- Characterization of semicontinuous pairings

## Abstract

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.