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

Abstract

We study pointwise convergence properties of weakly* converging sequences {ui} 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 |Du|. Furthermore, we discuss the optimality of these results.