Subgraphs of $\mathrm{BV}$ functions on $\mathrm{RCD}$ spaces

Abstract

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 $\mathrm{RCD}(K,N)$ 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 $\mathrm{BV}$ 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 $\mathrm{BV}$ function $f$ with respect to the polar vector of $f$, and we prove change-of-variable formulas.

20 pages. arXiv admin note: text overlap with arXiv:2204.04921