Strict and pointwise convergence of BV functions in metric spaces

Abstract

In the setting of a metric space $X$ equipped with a doubling measure that supports a Poincaré inequality, we show that if $u_i\to u$ strictly in $BV(X)$, i.e. if $u_i\to u$ in $L^1(X)$ and $\Vert D{u}_i\Vert (X)\to \Vert Du\Vert (X)$, then for a subsequence (not relabeled) we have ${\tilde{u}}_i(x)\to \tilde{u}(x)$ for $\mathcal{H}$-almost every $x\in X\setminus S_u$.

arXiv admin note: text overlap with arXiv:1612.06286