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Let $f:\Omega \subset \mathbb{R}^N \to \mathbb{R}^N$ be a vector field such that $f \in BV(\Omega)$.

Suppose that $\mathrm{div} f$ is absolutely continuous with respect to the Lebesgue measure and that its density is in $L^\infty$.

What does this imply about the derivative of $f$? For example, about its Cantor part?


Update 1: Is it true that $\mathrm{div}\, f$ is equal to the trace of $D_S f$ (that is to the trace of the rank-one matrix $M$ such that $D_S f = M|D_S f|$)? Why?

Update 2: What is the form of the rank-one matrix $M$ such that $D_S f = M|D_S f|$? That is, what are its entries in general? What are its entries in the case $\mathrm{div} \, f$ is absolutely continuous with respect to the Lebesgue measure?

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According to G. Alberti (Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), no. 2, 239–274), the singular part of $Df$ is a rank-one measure $D_Sf$. This is true for every BV vector field. When ${\rm div}\,f$ is a.c. with respect to the Lebesgue measure, then the trace of $D_Sf$ vanishes. However, I don't see what kind of information is conveyed by the boundedness of the divergence.

Edit. By the way, ${\rm div}\,f$ is not equal to ${\rm Tr}\,D_Sf$ in general. The correct statement is that the singular part of the measure ${\rm div}\,f$ equals ${\rm Tr}\,D_Sf$.

With your notation, $M$ is rank-one $|D_Sf|$-almost everywhere. It can be written $ab^T$ for $|D_Sf|$-measurable vector fields $a,b$. When ${\rm div}\,f$ is Lebesgue-a.c., then $a\cdot b=0$, $|D_Sf|$-almost everywhere.

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  • $\begingroup$ I see. That's very helpful. Where can I find a proof of those statements? For clarity, how do you write $ab^T$ as a matrix? And how do you write it if $a \cdot b = 0$? $\endgroup$ – Riku Apr 11 at 13:02
  • $\begingroup$ @Riku. $ab^T$ is naturally the ran-one matrix with entries $a_ib_j$, and $a\cdot b$ is the standard notation for scalar product. $\endgroup$ – Denis Serre Apr 11 at 14:11
  • $\begingroup$ Thank you for the clarification. Is it possible to obtain what $a_ib_j$ are in terms of $f$? Could you recommend a reference where these statements on the trace of $D_Sf$ are proved? $\endgroup$ – Riku Apr 11 at 14:13
  • $\begingroup$ Also, I've asked a related question here mathoverflow.net/questions/327789/… that you might be interested in. $\endgroup$ – Riku Apr 11 at 15:38

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