Spatial BV and distributional total variation

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open and let $u\in L^1(\mathrm{\Omega };\mathbb{R})$, where elements of $L^1$ are almost-everywhere equivalence classes. For each $i\in \{1,\dots ,n\}$, its distributional derivative $D_{i}u$ acts on $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ by $⟨D_{i}u,\phi ⟩:=-{\int }_{\mathrm{\Omega }}u{\mathrm{\partial }}_i\phi dx\mathrm{.}$ The function $u$ is in $BV(\mathrm{\Omega })$ when each $D_{i}u$ is represented by a finite signed Radon measure and the resulting vector-valued measure $Du=(D_{1}u,\dots ,D_{n}u)$ has finite total variation. Here a finite vector-valued Radon measure means that its partition variation is a finite Radon measure. For a Borel set $A\subseteq \mathrm{\Omega }$, define $\mathrm{\mid }Du\mathrm{\mid }(A):=\sup \{{\sum }_{j=1}^{\mathrm{\infty }}\mathrm{\mid }Du(A_j)\mathrm{\mid }:(A_j{)}_{j\ge 1}\text{ is a Borel partition of }A\},$ using the Euclidean norm on each vector value. Define the spatial total variation by $TV(u;\mathrm{\Omega }):=\mathrm{\mid }Du\mathrm{\mid }(\mathrm{\Omega })$. A Lebesgue-measurable set $E\subseteq \mathrm{\Omega }$ of finite measure has finite perimeter relative to $\mathrm{\Omega }$ when ${\chi }_E\in BV(\mathrm{\Omega })$; its relative perimeter is $P(E;\mathrm{\Omega }):=TV({\chi }_E;\mathrm{\Omega })$.

The distributional derivative uses the negative transpose sign on compactly supported tests.

An L1 element is an almost-everywhere equivalence class.

Radon measures are finite on compact sets and regular.

For signed or complex measures, regularity is carried by total variation.

A finite signed Radon measure representing a distribution on smooth compactly supported tests is unique.

The partition variation of a finite vector of signed Radon measures is a finite positive Radon measure.