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

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.