Measures in the dual of $BV$: perimeter bounds and relations with divergence-measure fields

TL;DR

This paper investigates measures in the dual of BV space, focusing on perimeter bounds, their relation to divergence-measure fields, and approximation techniques, with implications for weak solutions to curvature equations.

Contribution

It refines the understanding of perimeter bounds in the dual of BV, explores their connection with divergence-measure fields, and develops new approximation methods for BV functions and pairings.

Findings

Refined perimeter bounds for measures in the dual of BV.

Established stability of perimeter bounds under approximation.

Developed new bounds and approximation techniques for BV functions and pairings.

Abstract

We analyze some properties of the measures in the dual of the space $BV$, by considering (signed) Radon measures satisfying a perimeter bound condition, which means that the absolute value of the measure of a set is controlled by the perimeter of the set itself, and whose total variations also belong to the dual of $BV$. We exploit and refine the results of [25](Phuc, Torres 2017), in particular exploring the relation with divergence-measure fields and proving the stability of the perimeter bound from sets to $BV$ functions under a suitable approximation of the given measure. As an important tool, we obtain a refinement of Anzellotti-Giaquinta approximation for $BV$ functions, which is of separate interest in itself and, in the context of Anzellotti's pairing theory for divergence-measure fields, implies a new way of approximating $\lambda$-pairings, as well as new bounds for their total variation. These results are also relevant due to their application in the study of weak solutions to the non-parametric prescribed mean curvature equation with measure data, which is explored in a subsequent work.