This question is about some heuristics and graphs of BV functions.
In 1-dimensional setting, two key examples of $BV$ functions $u: \mathbb R \to \mathbb R$ are
the Heaviside function, whose derivative is the Dirac delta concentrated at $0$. It is an example of $BV$ function such that $Du = D^{jump}u$.
the Cantor function, whose derivative is the Hausdorff measure $\log_2 3$ restricted to the Cantor set. It is an example of BV function such that $Du = D^{cantor}u$.
The graphs of these functions and an intuition on the structure of their derivatives are quite straightforward. And actually, virtually all examples of BV functions I've seen in textbooks are 1-dimensional.
However, in the multi-dimensional setting I'm having some difficulties forming an intuition on these matters.
My question has a numerical component and a theoretical component.
I'd like to see two nice explicit examples of functions $u:\mathbb{R}^2 \to \mathbb{R}^2$ which are $BV$ and have respectively $Du=D^{jump}u$ and $Du=D^{cantor}u$.
using computer software like Mathematica or Matlab, I'd like to see the graph of such functions and a plot of their derivatives (so as to also see what the function at the jump set looks like the heuristic meaning of Alberti rank-one theorem on the "direction" of singularities, discussed in two other questions that appeared on MathOverflow Meaning of Alberti rank-one theorem and Alberti rank one theorem and a blow-up argument). I'd also be interested in seeing a representation of the level sets of $u$.
