Functions with Bounded Variation

This chapter is dedicated to a review of the space of functions with bounded variation in 1 and 2 dimensions. Indeed, BV  is the space in which entropy solutions live if the initial data belong to BV . We give several examples and prove some qualitative results and some compactness theorems that play a key role in the existence results. The proof and the definitions require only the knowledge of some basic measure theory because we do not consider the general n-dimensional case.