Traces of BV Functions

Abstract

Since we are regarding BV(Ω) as a subset of L ¹(Ω) we consider equivalence classes of functions rather than the functions themselves. Thus it makes no sense to talk of the value of a BV function on a set of measure zero since we may change the values of the function on such a set without changing its equivalence class. However it is important to be able to talk about the value of a BV function on the boundary of a set even though such a boundary may have measure zero. Obviously such a notion of values on a set of measure zero must take into account the value of the function on surrounding sets rather than just the set itself. It is the aim of this chapter to give a rigorous and meaningful definition of the trace of a BV function on the boundary of the set and then to develop some of the properties of the trace.