Quasiopen Sets, Bounded Variation and Lower Semicontinuity in Metric Spaces

摘要

In the setting of a complete metric space that is equipped with a doubling measure and supports a Poincare inequality, we show that the total variation of functions of bounded variation is lower semicontinuous with respect to L-1-convergence in every 1-quasiopen set. To achieve this, we first prove a new characterization of the total variation in 1-quasiopen sets. Then we utilize the lower semicontinuity to show that the variation measures of a sequence of functions of bounded variation converging in the strict sense are uniformly absolutely continuous with respect to the 1-capacity.