Examples of BV functions $u:\mathbb{R}^2 \to \mathbb{R}^2$ with singular derivative What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}^2$ with singular derivative?
More precisely, I'd like to see an example of 


*

*a function 
$$u_1 \in BV(\mathbb R^2; \mathbb R^2)$$
with only jump part in the derivative $$Du_1 = D^{jump} u_1$$

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

A related more general question is on MathOverflow.
 A: You can turn a one-dimensional BV functions $f$ into higher-dimensional examples by defining a radial function $u(x) = f(\|x\|)$. E.g., if $f(x)=-1$ for $x \le 1$ and $f(x)=0$ for $x>1$, then $u(x) = f(\|x\|)$ as a function from $\mathbb{R}^n$ to $\mathbb{R}$ has as distributional derivative the surface measure on the unit sphere, by the divergence theorem, so it is a BV function with $Du$ consisting only of a "jump" part. Explicitly, if $\phi:\mathbb{R}^2 \to \mathbb{R}^2$ is a smooth test function with compact support, then
$$
  \int_{\mathbb{R}^2} \phi \cdot \nabla u= -\int_{\mathbb{R}^2} u \,\, \textrm{div}\, \phi = \int_{\|x\|\le 1} \textrm{div}\, \phi = \int_{\|x\|=1} \phi \cdot n \, dS
$$
where the first integral is "formal" in the sense that $\nabla u$ does not exist as a function, with the precise meaning given by the second integral (using integration by parts/Green's formula over a large disk), and the last equality is the divergence theorem, with $n$ the normal vector and $dS$ the boundary measure, in this case 1-dimensional length measure on the circle. Since this is true for all test functions, we get that $\nabla u = n \, dS$ in the distributional sense.
A similar example with the Cantor staircase will give you a function whose derivative only has a singularly continuous ("Cantor") part.
