Functions of Bounded Variation and Sets of Finite Perimeter

The aim of this short course is to introduce the notion of pairing between a bounded divergence-measure vector field and the measure derivative of a function of bounded variation. To this end, we begin by recalling the basic properties of functions of bounded variation and of sets of finite perimeter, which are fundamental for understanding the subsequent material. We then discuss the notion of weak normal trace of a bounded divergence-measure vector field along a countably (N−1)-rectifiable set. Next, we introduce pairing measures and examine their main properties, with particular emphasis on their representation in terms of weak normal traces. We show how these pairing measures can be used to derive a generalized Gauss–Green formula. Finally, we discuss some generalizations, namely, the so-called lambda-pairings and nonlinear pairings, together with applications to semicontinuity and the relaxation of functionals in BV.