# 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.

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.

• This is interesting. 1. Could you please add a picture of the radial $2$-dimensional version of Cantor Staircase (with Mathematica or Matlab)? 2. Do you have any other genuinely 2-dimensional examples in mind?
– Riku
Apr 26 '19 at 16:51
• Also, I'm not sure that $f(\|x\|)$ satisfies Alberti rank-one theorem. What would the distributional derivative of $f(\|x\|)$ be?
– Riku
Apr 26 '19 at 18:05
• @Riku: Just take a picture of the 1-dimensional Cantor staircase and rotate it about the $y$-axis... The distributional gradient of $u$ is given by the flux through the unit sphere, and it is trivially of rank one here, because the codomain is one-dimensional. (I realize you asked for an example from $\mathbb{R}^2$ to $\mathbb{R}^2$, but you can just compose the given $u$ with a trivial embedding of the real line into the plane, or more generally some Lipschitz embedding.) Apr 26 '19 at 20:51
• What do you mean when you say that "The distributional gradient of u is given by the flux through the unit sphere"?
– Riku
Apr 26 '19 at 20:54
• Also, the singularity appears to be on concentric circles around the origin, which does not seem "unidirectional" in the sense of Alberti rank-one theorem. What am I missing?
– Riku
Apr 26 '19 at 20:58