# Concrete example of BV function $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative

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).

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.
• 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?
• 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?
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.
• More generally, I wonder what the possibilities are for the support of $Du$ given that it has only jump parts.. Jan 19 '20 at 8:58