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By Alberti rank-one theorem, could it be possible to reduce the study of a function $u \in BV(\mathbb{R}^N, \mathbb{R}^N)$ to the study of a function $\tilde{u} \in BV(\mathbb{R}^2, \mathbb{R}^2)$? At least as far as the derivative is concerned?

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    $\begingroup$ You should be more specific about what you mean by "reduction". $\endgroup$
    – Bazin
    Commented Apr 12, 2019 at 9:49
  • $\begingroup$ @Bazin I don't know precisely, I'd like to know about anything that can be done. I was thinking maybe one direction given by the "singular one" given by Alberti rank one and the other one $\mathbb{R}^{N-1}$ the orthogonal. $\endgroup$
    – user123457
    Commented Apr 12, 2019 at 17:59
  • $\begingroup$ As written in my answer on Mathoverflow, taking a $𝐡𝑉$ vector field on $\mathbb R^n$ with absolutely continuous divergence amounts essentially to deal with $𝑋=\sum_{1≀𝑗≀n}π‘Ž_j(π‘₯β€²,π‘₯_n)\frac{\partial}{\partial x_j}$ where $π‘₯β€²\in \mathbb R^{n-1}$ and $\frac{\partial a_1}{\partial x_n}$ Radon measure whereas all other entries of $𝐷𝑋$ are absolutely continuous. Although it is not sufficient, it means that a good idea is to start with $π‘Ž_1(π‘₯_1,π‘₯_2)\frac{\partial}{\partial x_1}$ in $\mathbb R^2.$ $\endgroup$
    – Bazin
    Commented Apr 15, 2019 at 13:00

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