Strict and pointwise convergence of BV functions in metric spaces (Q2014076)

Let \((X,d,\mu)\) be a metric space equipped with a doubling measure \(\mu\) that supports a \((1,1)\)-Poincaré inequality and let \(\mathcal{H}\) denote the codimension one Hausdorff measure in \(\mathcal{P}(X)\). Assume that \(\Omega\subset X\) is an open set, \((u_i)_i\subset \mathrm{BV}(\Omega)\) is a sequence which converges strictly in \(\mathrm{BV}(\Omega)\) to a function \(u\), i.e., \(u_i\to u\) in \(L^1(\Omega)\) and \(||Du_i||(\Omega)\to ||Du||(\Omega)\). Then there exists a subsequence (not relabeled) such that \(\tilde{u}_i(x)\to\tilde{u}(x)\) for \(\mathcal{H}\)-almost every \(x\in \Omega\setminus S_u\), where \(S_u\) is the jump set of \(u\). Moreover, for any compact set \(K\subset\Omega\setminus S_u\) and \(\varepsilon>0\) there exists an open set \(U\subset X\) with \(\mathrm{Cap}_1(U)<\varepsilon\) such that \(\tilde{u}_i\to\tilde{u}\) uniformly in \(K\setminus U\). The results are new even in the Euclidean setting. The main tool used in the proofs is a boxing-type argument, see [\textit{J. Kinnunen} et al., Indiana Univ. Math. J. 57, No. 1, 401--430 (2008; Zbl 1146.46018)]. Examples showing that the obtained results are optimal are given at the end of the paper.