Quasiopen sets, bounded variation and lower semicontinuity in metric spaces

TL;DR

This paper establishes the lower semicontinuity of total variation for functions of bounded variation in metric spaces with doubling measures and Poincaré inequalities, focusing on 1-quasiopen sets and their properties.

Contribution

It provides a new characterization of total variation in 1-quasiopen sets and demonstrates lower semicontinuity and measure convergence properties in this setting.

Findings

Total variation is lower semicontinuous with respect to $L^1$-convergence in 1-quasiopen sets.

Variation measures of BV functions converge and are uniformly absolutely continuous w.r.t. 1-capacity.

New characterization of total variation in 1-quasiopen sets.

Abstract

In the setting of a metric space that is equipped with a doubling measure and supports a Poincar\'e 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.