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

Giovanni E. Comi, Gian Paolo Leonardi

TL;DR

The work characterizes the dual of $BV(\Omega)$ via a perimeter-bound condition $|\mu(E^{1}\cap\Omega)| \le L P(E)$ and proves that $\mu \in BV(\Omega)^*$ if and only if $\mu$ obeys this bound and $\mu=F$ for some $F\in \mathcal{DM}^{\infty}(\Omega)$, linking $BV^*$ to divergence-measure fields. It introduces admissible measures with $|\mu|\in BV(\Omega)^*$ and develops a refined $\lambda$-approximation of BV functions that ensures convergence of $u^{\lambda}$ and the area functional, enabling sharp bounds for $\lambda$-pairings and robust representations for integration by parts. A key technical achievement is the enhanced Anzellotti–Giaquinta approximation that preserves $L^{\infty}$ bounds and traces, leading to stability results and a density approximation framework that extends to measure data problems. These results underpin a Gamma-convergence approach to the prescribed mean curvature equation with measure data and provide a rigorous bridge between BV-duality, divergence-measure fields, and geometric measure-theoretic tools, with implications for nonlinear PDEs with measure data.

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 $λ$-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.

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

TL;DR

The work characterizes the dual of via a perimeter-bound condition and proves that if and only if obeys this bound and for some , linking to divergence-measure fields. It introduces admissible measures with and develops a refined -approximation of BV functions that ensures convergence of and the area functional, enabling sharp bounds for -pairings and robust representations for integration by parts. A key technical achievement is the enhanced Anzellotti–Giaquinta approximation that preserves bounds and traces, leading to stability results and a density approximation framework that extends to measure data problems. These results underpin a Gamma-convergence approach to the prescribed mean curvature equation with measure data and provide a rigorous bridge between BV-duality, divergence-measure fields, and geometric measure-theoretic tools, with implications for nonlinear PDEs with measure data.

Abstract

We analyze some properties of the measures in the dual of the space , 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 . 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 functions under a suitable approximation of the given measure. As an important tool, we obtain a refinement of Anzellotti-Giaquinta approximation for 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 -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.
Paper Structure (7 sections, 22 theorems, 230 equations)

This paper contains 7 sections, 22 theorems, 230 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Functions of bounded variation
  4. Divergence-measure fields and $\lambda$-pairings
  5. Measures in the dual of $BV$
  6. $\lambda$-approximation
  7. Perimeter bounds and admissible measures

Key Result

Proposition 2.1

Let $u \in BV_{\rm loc}(\Omega)$, $\lambda:\Omega\to [0,1]$ be a Borel function and $N > 0$. For all $x \in \Omega \setminus (S_u \setminus J_u)$ we set Then $M[u,\lambda](x)$ is well defined for ${\mathcal{H}}^{n - 1}$-a.e. $x \in \Omega$, and it satisfies Moreover, we have for all $x \in \Omega$ such that $- N \le u^{-}(x) \le u^{+}(x) \le N$, which in turn implies $T_{N}(u)^{\lambda}(x) \to

Theorems & Definitions (62)

  • Definition 1.1
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Proposition 2.7
  • ...and 52 more