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 $u_i^{\ast }(x)\to u^{\ast }(x)$ 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 $|Du|$. Furthermore, we discuss the optimality of these results.

arXiv comment22 pages, comments are welcome