Quasiopen sets, bounded variation and lower semicontinuity in metric spaces

2017/03/14 by Panu Lahti, Lahti, Panu
Mathematics · #26B30 #30L99 #31E05 #FOS: Mathematics #Metric Geometry (math.MG) #math.MG #msc:26B30 #msc:30L99 #msc:31E05

paper · pdf · doi:10.48550/arxiv.1703.04675

arxiv created 2017/03/14 · arxiv updated 2017/03/16

Abstract

In the setting of a 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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