Functions of Bounded Variation

Abstract

A function of bounded variation of one variable can be characterized as an integrable function whose derivative in the sense of distributions is a signed measure with finite total variation. This chapter is directed to the multivariate analog of these functions, namely the class of L1functions whose partial derivatives are measures in the sense of distributions. Just as absolutely continuous functions form a subclass of BV functions, so it is that Sobolev functions are contained within the class of BV functions of several variables. While functions of bounded variation of one variable have a relatively simple structure that is easy to expose, the multivariate theory produces a rich and beautiful structure that draws heavily from geometric measure theory. An interesting and important aspect of the theory is the analysis of sets whose characteristic functions are BV (called sets of finite perimeter). These sets have applications in a variety of settings because of their generality and utility. For example, they include the class of Lipschitz domains and the fact that the Gauss-Green theorem is valid for them underscores their usefulness. One of our main objectives is to establish Poincare-type inequalities for functions of bounded variation in a context similar to that developed in Chapter 4 for Sobolev functions. This will require an analysis of the structure of BV functions including the notion of trace on the boundary of an open set.