Measures in the dual of $BV$: perimeter bounds and relations with divergence-measure fields
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| Format: | Preprint |
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2024
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| author | Comi, Giovanni E. Leonardi, Gian Paolo |
| author_facet | Comi, Giovanni E. Leonardi, Gian Paolo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_06224 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Measures in the dual of $BV$: perimeter bounds and relations with divergence-measure fields Comi, Giovanni E. Leonardi, Gian Paolo Analysis of PDEs Functional Analysis Primary: 26B30. Secondary: 26B20, 46E27 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. |
| title | Measures in the dual of $BV$: perimeter bounds and relations with divergence-measure fields |
| topic | Analysis of PDEs Functional Analysis Primary: 26B30. Secondary: 26B20, 46E27 |
| url | https://arxiv.org/abs/2407.06224 |