Functions of bounded variation and the AM-modulus in Rn. (December 2018) - Honzlová Exnerová, Vendula; Malý, Jan; Martio, Olli
Abstract: Moduli of path families are widely used to study Sobolev functions. Similarly, the recently introduced approximation ( A M -) modulus is helpful in the theory of functions of bounded variation ( B V ) in R n (Martio, 2016). We continue this direction of research. Let Γ E be the family of all paths which meet E ⊂ R n . We introduce the outer measure E ↦ A M ( Γ E ) and compare it with other ( n − 1 ) -dimensional measures. In particular, we show that A M ( Γ E ) = 2 H n − 1 ( E ) whenever E lies on a countably ( n − 1 ) -rectifiable set. Further, we study functions which have bounded variation on A M -a.e. path and we relate these functions to the classical B V functions which have only bounded essential variation on A M -a.e. path. We also characterize sets E of finite perimeter in terms of the A M -modulus of two path families naturally associated with E .