A chain rule formula in BV(Omega) and applications to lower semicontinuity

A new lower semicontinuity theorem for integral functionals with linear growth is proved. This result is obtained without assuming any continuity of the energy density f with respect to the spatial variable x, but by requiring only a BV-dependence. The proof is based on a new chain rule formula in the space BV which does not seem to be contained in any of the similar results known in the literature and which could be of independent interest