2
$\begingroup$

Motivated by my previous question Alberti rank-one theorem and irregular jump discontinuities, I'd like to ask the following:

Is there a function $u \in BV_{loc}(\mathbb{R}^2)$ such that $Du$ is an $s$-dimensional Hausdorff measure restricted to the Koch curve?

In particular, consider the square $[0,1] \times [0,1]$ and the Koch curve with end points $\{0,1\}$.

Does the characteristic function of the part of the square below the curve satisfy the requirements of the question?

Improve this question
$\endgroup$
10
  • $\begingroup$ Is $s$ the Hausdorff dimension of the Koch curve ($s={\log4\over\log3}$) ? $\endgroup$
    – Pietro Majer
    Commented Apr 24, 2019 at 13:51
  • $\begingroup$ @PietroMajer Yes. $\endgroup$
    – Riku
    Commented Apr 24, 2019 at 13:55
  • $\begingroup$ Since $Du$ is zero outside of the Koch curve, $u$ should be an indicator of the set $E$ whose boundary is the Koch curve. Then $u\in BV$ iff $E$ has finite perimeter, which does not hold since $s>1$. $\endgroup$
    – Skeeve
    Commented Apr 24, 2019 at 23:21
  • $\begingroup$ @Skeeve That's right and very unfortunate because it makes the question mathoverflow.net/questions/328779/…, which that motivates this one much harder for me. $\endgroup$
    – Riku
    Commented Apr 24, 2019 at 23:23
  • $\begingroup$ @Skeeve On the other hand, is the characteristic function of the undergraph of the koch curve BV? $\endgroup$
    – Riku
    Commented Apr 24, 2019 at 23:29

0

Reset to default

You must log in to answer this question.