A note on the weak* and pointwise convergence of BV functions. (December 2022)

• Record Type: Journal Article
• Authors: Beck, Lisa; Lahti, Panu
• Is Part Of: Nonlinear analysis. Volume 225(2022)
• Publication Date: 2022-12
• Subjects: primary 26B30 28A20 31C40 -- secondary 28A78; Functions of bounded variation -- Weak star convergence -- Pointwise convergence -- Variation measure -- Cantor part -- Hausdorff dimension; 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.2022.113028
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study pointwise convergence properties of weakly* convergent sequences { u i } i ∈ N in BV ( R n ) . We show that, after passage to a suitable subsequence (not relabeled), we have pointwise convergence u i ∗ ( x ) → u ∗ ( x ) of the precise representatives for all x ∈ R n ∖ E, where the exceptional set E ⊂ R n has on the one hand Hausdorff dimension at most n − 1, and is on the other hand also negligible with respect to the Cantor part of | D u | . Furthermore, we discuss the optimality of these results.