In 1-dimensional setting, two key examples of $BV$ functions (of bounded variation) $u: \mathbb R \to \mathbb R$ are the Heaviside function, whose derivative is the Dirac delta concentrated at $0$. It is an example of $BV$ function such that $Du = D^{jump}u$ and the Cantor staircase function, whose derivative is the $\log_3(2)$-dimensional Hausdorff measure restricted to the Cantor set. It is an example of BV function such that $Du = D^{cantor}u$. It is straightforward to plot these functions using Mathematica.

However, I'd like to see the graph of two functions $u:\mathbb R^2 \to \mathbb R^2$ which are $BV$ and have respectively $Du=D^{jump}u$ and $Du=D^{cantor}u$ (that is, only jump or Cantor part respectively in the derivative).

How can I use Mathematica to plot such functions and get a representation of their derivatives?

One motivation for this is to see what $BV$ in multiple space dimensions look like at the jump set and to check the heuristic meaning of Alberti rank-one theorem on the "direction" of singularities. For more context and motivation, see a related question on MathOverflow.

  • $\begingroup$ @HenrikSchumacher Yes, thank you for correction. $\endgroup$ – Riku Apr 21 at 14:59

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