The classical Gauss–Green theorem on open sets with C1-boundary plays a fundamental role in the theory of sets of finite perimeter. We shall build its proof on a nice application of the area formula to codimension one graphs (Section 9.1), and then generalize it to the case of open sets whose boundaries fail to be of class C1 due to the presence of an Hn–1-negligible set (Section 9.3). Throughout this chapter we use Notation 4.
9.1 Area of a graph of codimension one
Given u : n–1 → and G ⊂ n–1, we define the graph of u