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What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative?

More precisely, I'd like to see an example (and a plot using Mathematica or Matlab) of

  • a function $$u_1 \in BV(\mathbb R^2; \mathbb R)$$ with only jump part in the derivative $$Du_1 = D^{jump} u_1$$
  • and of a function with only Cantor part in the derivative: $$u_2 \in BV(\mathbb R^2; \mathbb R)$$ with $$Du_2 = D^{cantor} u_2$$

A related more general question is Heuristic and graphic representation of BV functions and their singularities


Clearly one could take a one dimensional example $f \in BV(\mathbb R)$ and then consider $g(x_1,x_2) := f(x_1)$. However, I'd like to see a "genuinely" two-dimensional example (if it exists).

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The absolutely continuous part and the jump part of the derivative are easy to understand: taking $\mathbf 1_{\mathbf R_+}(x_1)$ provides a $BV$ function with jump at $x_1=0$. As far as the Cantor part is concerned, you just have to think about the Cantor measure, namely the (distribution) derivative of the "devil" staircase: you know certainly the Cantor set $K$ which appears as $$ K=\cap_{n\ge 0}K_n, $$ with $K_0=[0,1]$, $K_1=[0,1/3]\cup[2/3,1]$, $K_n$ is the union of $2^n$ compact intervals $I_{n,l}$with length $3^{-n}$ and $$K_{n+1}=\cup_{1\le l\le 2^n}\bigl(I_{n,l}\backslash {\text{its open middle third}}\bigr).$$ Then it is easy to see that the probability measures $\mathbf 1_{K_n}/\vert K_n\vert$ converge weakly toward the so-called Cantor measure which is singular with respect to the Lebesgue measure and is diffuse (no atoms). An antiderivative of the Cantor measure belongs indeed to $BV$ with a singular derivative without jumps.

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  • $\begingroup$ Thank you. In your example you took $g(x_1,x_2) = f(x_1)$ and $f$ the Cantor staircase function. However, I'd like to see a "genuinely" two-dimensional example. Is it possible to build a Cantor like function in 2d? $\endgroup$ – Riku Apr 23 '19 at 21:18
  • $\begingroup$ Or is it fair to say that if $u:\mathbb R^2 \to \mathbb R$ is a BV function then $D^{cantor}u$ must be concentrated on a $1$-dimensional Cantor set? $\endgroup$ – Riku Apr 25 '19 at 0:29
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You could take $u$ to be the indicator function of a square. Then the jump part will be supported on the boundary of the square.

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  • $\begingroup$ More generally, I wonder what the possibilities are for the support of $Du$ given that it has only jump parts.. $\endgroup$ – James Baxter Jan 19 at 8:58

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