Calculus of Variations and Geometric Measure Theory

G. Antonelli - C. Brena - E. Pasqualetto

Subgraphs of $\rm BV$ functions on $\rm RCD$ spaces

created by pasqualetto on 02 Sep 2022



Inserted: 2 sep 2022

Year: 2022

ArXiv: 2209.00645 PDF


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.