On L¹-Lower Semicontinuity in BV

A new lower semicontinuity theorem for integral functionals defined on the space BV of functions of bounded variation is proved. This result is obtained by replacing the continuity of the energy density f with respect to the space variable x with a weak differentiability assumption. The proof is based on a new integration by parts 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