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

Giovanni E. Comi; Gian Paolo Leonardi

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arXiv · 2407.06224

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

Giovanni E. Comi·Gian Paolo Leonardi

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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.

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Giovanni E. Comi, Gian Paolo Leonardi. 2025-01-08. Measures in the dual of $BV$: perimeter bounds and relations with divergence-measure fields. https://arxiv.org/abs/2407.06224

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