Extensions and traces of functions of bounded variation on metric spaces

TL;DR

This paper studies how functions of bounded variation can be extended and analyzed on metric spaces with certain geometric properties, providing new insights into boundary traces and jump set behavior.

Contribution

It extends the theory of BV functions to metric spaces, establishing new results on extensions, boundary traces, and jump set behavior under specific conditions.

Findings

BV functions can be extended from bounded uniform domains to entire metric spaces.

New results on the behavior of BV functions in their jump sets.

Boundary traces of BV functions are characterized under locality conditions.

Abstract

In the setting of a metric space equipped with a doubling measure and supporting a Poincar\'e inequality, and based on results by Bj\"orn and Shanmugalingam (2007), we show that functions of bounded variation can be extended from any bounded uniform domain to the whole space. Closely related to extensions is the concept of boundary traces, which have previously been studied by Hakkarainen et al. (2014). On spaces that satisfy a suitable locality condition for sets of finite perimeter, we establish some basic results for the traces of functions of bounded variation. Our analysis of traces also produces novel results on the behavior of functions of bounded variation in their jump sets.