Lower Semicontinuity of Quasi-convex Bulk Energies in SBV and Integral Representation in Dimension Reduction

A result of Larsen concerning the structure of the approximate gradient of certain sequences of functions with Bounded Variation is used to present a short proof of Ambrosio’s lower semicontinuity theorem for quasiconvex bulk energies in SBV . It enables to generalize to the SBV setting the decomposition lemma for scaled gradients in dimension reduction and also to show that, from the point of view of bulk energies, SBV dimensional reduction problems can be reduced to analogue ones in the Sobolev spaces framework.