A note on the weak* and pointwise convergence of BV functions

TL;DR

This paper investigates the pointwise convergence of weak* converging sequences in BV spaces, establishing conditions under which convergence of precise representatives occurs outside a small exceptional set.

Contribution

It provides new results on the pointwise convergence of BV functions under weak* convergence, including the size and nature of the exceptional set where convergence may fail.

Findings

Pointwise convergence holds outside a set of Hausdorff dimension at most n-1.

The exceptional set is negligible with respect to the Cantor part of the derivative measure.

Results are shown to be optimal in terms of the size of the exceptional set.

Abstract

We study pointwise convergence properties of weakly* converging sequences $\{u_i\}_{i \in {\mathbb N}}$ in $\mathrm{BV}({\mathbb R}^n)$. We show that, after passage to a suitable subsequence (not relabeled), we have pointwise convergence $u_i^*(x)\to u^*(x)$ of the precise representatives for all $x\in {\mathbb R}^n \setminus E$, where the exceptional set $E \subset {\mathbb R}^n$ 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.