Inserted: 25 mar 2013
Last Updated: 24 jul 2015
Journal: Ann. Inst. H. Poincaré Anal. Non Linéaire
(2015), no. 3, .
We give a notion of $BV$ function on an oriented manifold where a volume form and a family of lower semicontinuous quadratic forms $G_p: T_pM \to [0,\infty]$ are given. Using this notion, we generalize the structure theorem for $BV$ functions that holds in the Euclidean case. When we consider sub-Riemannian manifolds, our definition coincide with the one given in the more general context of metric measure spaces which are doubling and support a Poincar\'e inequality. We then focus on finite perimeter sets, i.e., sets whose characteristic function is $BV$, in sub-Riemannian manifolds. Under an assumption on the nilpotent approximation, we prove a blowup theorem, generalizing the one obtained for step-2 Carnot groups in 24.