Representation formulas for pairings between divergence-measure fields and $BV$ functions

TL;DR

This paper develops pointwise formulas for the pairing between divergence-measure fields and BV functions, extending previous representations and providing new limit-based descriptions of normal traces.

Contribution

It introduces new pointwise representation formulas for pairings between divergence-measure fields and BV functions, building on and extending prior work in the area.

Findings

Extended Anzellotti's representation formula using cylindrical averages.

Derived new representation in terms of limits of averages in half balls.

Provided a comprehensive framework for analyzing pairings in divergence-measure fields.

Abstract

The purpose of this paper is to find pointwise representation formulas for the density of the pairing between divergence-measure fields and BV functions, in this way continuing the research started in [17,20]. In particular, we extend a representation formula from an unpublished paper of Anzellotti [7] involving the limit of cylindrical averages for normal traces, and we exploit a result of [35] in order to derive another representation in terms of limits of averages in half balls.