# BV function with absolutely continuous divergence

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?

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.
• 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 $a \cdot b = 0$? – Riku Apr 11 '19 at 13:02
• @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. – Denis Serre Apr 11 '19 at 14:11
• 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? – Riku Apr 11 '19 at 14:13