Subgraphs of \(\rm BV\) functions on \(\rm RCD\) spaces

In this work we extend classical results for subgraphs of functions of bounded variation in \(\mathbb{R}^n\times\mathbb{R}\) to the setting of \(\mathsf{X}\times\mathbb{R}\), where \(\mathsf{X}\) is an \({\rm RCD}(K,N)\) metric measure space. In particular, we give the precise expression of the push...

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Bibliographic Details
Published in:arXiv.org 2022-09
Main Authors: Antonelli, Gioacchino, Brena, Camillo, Pasqualetto, Enrico
Format: Article
Language:English
Subjects:
Graph theory
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Summary:In this work we extend classical results for subgraphs of functions of bounded variation in \(\mathbb{R}^n\times\mathbb{R}\) to the setting of \(\mathsf{X}\times\mathbb{R}\), where \(\mathsf{X}\) is an \({\rm RCD}(K,N)\) metric measure space. In particular, we give the precise expression of the push-forward onto \(\mathsf{X}\) of the perimeter measure of the subgraph in \(\mathsf{X}\times\mathbb{R}\) of a \(\rm BV\) function on \(\mathsf{X}\). Moreover, in properly chosen good coordinates, we write the precise expression of the normal to the boundary of the subgraph of a \(\rm BV\) function \(f\) with respect to the polar vector of \(f\), and we prove change-of-variable formulas.
ISSN:2331-8422