A chain rule formula in BV and application to lower semicontinuity

Dv = B′(ũ)∇uLN +B′(ũ)Dcu+ (B(u)−B(u))νuH Ju , where ∇u is the absolutely continuous part of Du, Du is the Cantor part of Du and Ju is the jump set of u (for the definition of these and other relevant quantities, see Sect.2). A delicate issue about this formula concerns the meaning of the first two terms on the right hand side. In fact, in order to understand why they are well defined, one has to take into account that B′(t) exists for L-a.e. t and that, if E is an L-null set in IR, not only ∇u vanishes L -a.e. on ũ−1(E), but also |Dcu|(ũ−1(E)) = 0 (see [2, Theorem 3.92]). The difficulty of giving a correct meaning to the various parts in which the derivative of a BV function can be split is even greater when u is a vector field, a case where a chain rule formula has been proved by Ambrosio and Dal Maso in [1]. In particular, their result applies to the composition of a scalar BV function with a Lipschitz function B depending also on x, namely to the function B(x, u(x)), where B : Ω × IR → IR is Lipschitz. In many applications, however, B has the special form