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Let $u: \mathbb R^n \to \mathbb R$ be a $BV$ function with no jump part, i.e., writing $Du = D^a u + D^s u + D^j u$ for the decomposition of $Du$ into absolutely continuous, Cantor, and jump part respectively, we have $D^j u = 0$.

Since $u$ is a priori an equivalence class of functions $L_{loc}^1$, in the following question which is sensitive to variation on Lebesgue null sets, we use the precise representative of $u$, which is defined to be equal to the approximate limit of $u$, wherever it exists. This defines the function $u$ up to a set of $\mathcal H^{n-1}$ finite measure, which is enough for the purposes of the following:

Question: Does the weak derivative of $u$ vanish on level sets? In other words, is it true that $|Du|(u^{-1}(p)) = 0$ for all $p \in \mathbb R$?

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  • $\begingroup$ Can you confirm that this is what you intended to write? When $u$ is of class $C^1$ for example then $Du = D^a u = \nabla u \, \mathrm{d} \mathcal{H}^n$ with no jump or Cantor part, no? $\endgroup$ – Leo Moos Apr 5 at 23:40
  • $\begingroup$ Yes that’s right. For such a function, if the hypothesis is true we should have for each $r$ that $\ ∇ u = 0$ $\mathcal, H^n$-a.e. on the set $\{x \in \mathbb R \ | \ u(x) = r \}$. Although I think I want to change $Du(f^{-1} (p))$ to $|Du|(f^{-1} (p))$, which I will do so now. $\endgroup$ – Nate River Apr 5 at 23:50
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    $\begingroup$ How do you ensure that $|Du|(u^{-1}(p))$ is well defined? The standard-definition of $BV$ in $\mathbb{R}^n$ is as a subset of $L^1$, so a function is only defined up to Lebesgue zero-sets. This is enough to define $|D^au|(u^{-1}(p))$, but for the singular part one could take a representative with either $u = p$ on the support of $D^su$ or one with $u \neq p$, which would change the answer. $\endgroup$ – mlk Apr 6 at 12:54
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    $\begingroup$ Yes, but as stated in my previous comment, as long as there is any Cantor-part, there is always a representative for which the statement is false. I.e. you can set $u=p$ on the zero-set $\mathrm{supp} D^su$, and then $|Du|(u^{-1}(p)) \geq |D^su|(u^{-1}(p)) = |D^su|(\mathbb{R}^n) > 0$. $\endgroup$ – mlk Apr 10 at 9:13
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    $\begingroup$ Ah you’re right. The proper move here is to use its “precise representative” which is equal to its approximate limit everywhere except the jump part (which doesn’t exist in this case). I will update the post, thanks for pointing this out. $\endgroup$ – Nate River Apr 10 at 10:17
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This is about the univariate case ($n = 1$) only, but way too long for a comment.


Step 1: notation. Let $u$ be a continuous function of bounded variation, let $\mu = Du$ be the corresponding Radon measure, let $|\mu|$ denote the variation measure of $\mu$, and let $U$ be the distribution function of $|\mu|$: $$ U(x) = \begin{cases} |\mu|([0,x)) & \text{if } x \geqslant 0 \\ |\mu|([x, 0)) & \text{if } x < 0 \end{cases} $$ (so that $|\mu| = D U$). Let $S$ be the support of $\mu$, and let $S'$ be equal to $S$ right endpoints of every connected component of $S$ removed. Then $|\mu|$ assigns no measure to the complement of $S'$ (because it assigns no measure both to the complement of $S$ and to the countable set $S \setminus S'$), and $U$ is a one-to-one mapping of $S'$ onto some interval $I$. The notation would have been much simpler if we assumed that $S$ is an interval, but let us deal with the general case.

Define a measure $\nu$ on $I$ by the formula $$ \nu(B) = \mu(U^{-1}(B)) = \mu(U^{-1}(B) \cap S') . $$ Since $U : S' \to I$ is a bijection and $\mu(A) = \mu(A \cap S') = \nu(U(A))$, we have $$ \nu_\pm(B) = \mu_\pm(U^{-1}(B)) . $$ Therefore, $$ |\nu|(B) = |\mu|(U^{-1}(B)) = |B| $$ is the Lebesgue measure of $B$ (as long as $B$ is contained in $I$, of course). It follows that $\nu$ is absolutely continuous on $I$, with density function equal to $\pm 1$ almost everywhere.

Let $w(y) = u(U^{-1}(y))$, where $U^{-1}(y)$ is the unique point $x$ in $S'$ such that $U(x) = y$ (note, however, that since $v$ is constant on every connected component of the complement of $S$, any other choice of $x$ would work equally well). It is easy to check that $w$ is the distribution function of $\nu$ (that is, $\nu = D w$), so that $w'$ exists almost everywhere, and $|w'| = 1$ almost everywhere.

Clearly, for any $p$ we have $$|\mu|(u^{-1}(\{p\})) = |\mu|(u^{-1}(\{p\}) \cap S') = |\nu|(U(u^{-1}(\{p\}))) = |\nu|(w^{-1}(\{p\})) = |w^{-1}(\{p\})|.$$ Note that $w' = 0$ almost everywhere on $w^{-1}(\{p\})$ (indeed: if $w'(y)$ exists for some non-isolated point $y$ in $w^{-1}(\{p\})$, then $w'(y) = 0$). However, $|w'| = 1$ almost everywhere. This means that $w^{-1}(\{p\})$ has zero Lebesgue measure, and consequently $$ |\mu|(u^{-1}(\{p\})) = 0 ,$$ as desired.


I have no experience with multivariate functions of bounded variation, so I will stop here. Let me remark, however, that the multivariate case might follow from the univariate one if, for example, there is a bounded variation counterpart of the "absolutely continuous on lines" (ACL) characterisation of weakly differentiable functions.

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    $\begingroup$ That’s a good idea on the ACL part! I believe there is a “BVL” (bounded variation on lines) characterisation of BV functions, provided in Ambrosio’s book, Functions of Bounded Variation and Free Discontinuity Problems. I know where to take a look now. Thanks! $\endgroup$ – Nate River Apr 6 at 0:30

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