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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1$\begingroup$ You should be more specific about what you mean by "reduction". $\endgroup$– BazinCommented Apr 12, 2019 at 9:49
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$\begingroup$ @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. $\endgroup$– user123457Commented Apr 12, 2019 at 17:59
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$\begingroup$ As written in my answer on Mathoverflow, taking a $ðĩð$ vector field on $\mathbb R^n$ with absolutely continuous divergence amounts essentially to deal with $ð=\sum_{1âĪðâĪn}ð_j(ðĨâē,ðĨ_n)\frac{\partial}{\partial x_j}$ where $ðĨâē\in \mathbb R^{n-1}$ and $\frac{\partial a_1}{\partial x_n}$ Radon measure whereas all other entries of $ð·ð$ are absolutely continuous. Although it is not sufficient, it means that a good idea is to start with $ð_1(ðĨ_1,ðĨ_2)\frac{\partial}{\partial x_1}$ in $\mathbb R^2.$ $\endgroup$– BazinCommented Apr 15, 2019 at 13:00
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