Thanks a lot for your answer! I found rather surprising that very little is known on this topic, I did expect a huge literature on this. I will look through the papers you mentioned. A part from regularity, which seems out-of-reach, do you know whether some maximum principles are available in this setting? In the aforementioned paper by Bombieri & co. it is proved e.g. (using regularity) that if $U,V$ are open sets s.t. their boundaries of least area and $U,V$ coincide out of a compact then they are indeed equal $U=V$. Any ideas for a similar principle in anisotropic case? Thanks again.

+1, nice question! I am interested into the Eulerian side and I know Depauw's example. Could one do something similar to what @Bazin suggests also in that case?

Does your last expression adapt also to piecewise constant $\alpha$? Maybe that could be very useful to me... (I am not really in this framework but maybe I could get somehow close to it...) Thanks again!

Thanks for your precious effort and for writing the implicit expression of the solution for vanishing $\alpha$. I will try to work on it to see if I find something... Do you believe there is possibility of deriving some Gronwall-like estimates from the equation? Maybe writing it as an ODE is a good point... In any case, thanks again for your interest!

Thanks a lot for the comment. Sure it may be of interest, of course, though not fully resolutive (but maybe with some tricks I can work out some estimates also for the general $\alpha$). Could you please point me out some references? Thanks!

@Gro-Tsen Uh, I see. That's really interesting how nasty Hausdorff measures (and dimension) can be! If you want to write your comments as an answer I will mark it as the accepted answer. Thanks a lot!

@Gro-Tsen Thanks a lot for your comment, it makes perfectly sense. I think you are right: the Cantor set you constructed has still H-dimension $\alpha= \log 2 / \log 3$, nevertheless its staircase has not $\mathscr H^{\alpha}$ as derivative. Btw, one more question: the derivative of its staircase will still be some $\mathscr H^{\beta}$ (though not with $\beta = \alpha$), am I right? So in the end, is it true that the Cantor part of the derivative of a BV function is always (a.c. w.r.t. some) fractional Hausdorff measure? Thanks again.

Or maybe one could work with the sets $E_\alpha = \{x: \lim_{r \to 0} r^{-\alpha}|Du|(B^r(x)) > 0\}$... does this make sense? Thanks.

Oh you are right, that is true. So it is interesting to see whether this decomposition into $\alpha$-dimensional sets can be performed... the only thing that comes to my mind is coarea formula, but that does not seem to help... Thanks again.