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

TL;DR

This paper introduces a new pairing concept for measure vector fields with divergence measure and scalar functions, extending classical formulas and defining novel anisotropic perimeters, even for fractal boundaries.

Contribution

It develops a new pairing framework that applies to less regular functions and vector fields, generalizing Gauss-Green formulas and perimeter concepts beyond traditional BV functions.

Findings

Pairings preserve coarea formula and lower semicontinuity.

Gauss-Green formulas are extended to new function classes.

New anisotropic perimeters accommodate fractal boundaries.

Abstract

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.