A note on the weak* and pointwise convergence of BV functions. (December 2022) - Beck, Lisa; Lahti, Panu
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.