Strong approximation of sets of finite perimeter in metric spaces

TL;DR

This paper proves that in certain metric spaces, sets of finite perimeter can be closely approximated by sets with nearly matching topological and measure-theoretic boundaries, extending known Euclidean results.

Contribution

It introduces a new approximation result for sets of finite perimeter in metric spaces with doubling measures and Poincaré inequalities, even in Euclidean spaces.

Findings

Sets of finite perimeter can be approximated in BV norm by sets with nearly coinciding boundaries.

The approximation relies on a quasicontinuity property for BV functions.

The result is new even in Euclidean spaces.

Abstract

In the setting of a metric space equipped with a doubling measure that supports a Poincar\'e inequality, we show that any set of finite perimeter can be approximated in the BV norm by a set whose topological and measure theoretic boundaries almost coincide. This result appears to be new even in the Euclidean setting. The work relies on a quasicontinuity-type result for BV functions proved by Lahti and Shanmugalingam (2016).