In the setting of a complete metric space that is equipped with a doubling measure and supports a Poincaré inequality, we show that the total variation of functions of bounded variation is lower semicontinuous with respect to L1-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.
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The research was funded by a grant from the Finnish Cultural Foundation. The author wishes to thank Nageswari Shanmugalingam and two anonymous referees for giving helpful comments on the manuscript.
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Lahti, P. Quasiopen Sets, Bounded Variation and Lower Semicontinuity in Metric Spaces. Potential Anal 52, 321–337 (2020). https://doi.org/10.1007/s11118-018-9749-8
- Metric measure space
- Function of bounded variation
- Total variation
- Quasiopen set
- Lower semicontinuity
- Uniform absolute continuity
Mathematics Subject Classification (2010)