Functions of bounded variation and the AM-modulus in Rn. (December 2018)

• Record Type: Journal Article
• Authors: Honzlová Exnerová, Vendula; Malý, Jan; Martio, Olli
• Is Part Of: Nonlinear analysis. Volume 177(2018)Part B
• Page Start: 553
• Page End: 571
• Publication Date: 2018-12
• Subjects: Modulus of path family -- Functions of bounded variation; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2018.05.015
• Languages: English
• ISSNs: 0362-546X

Abstract

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 .