On rough traces of BV functions

TL;DR

This paper investigates boundary traces of BV functions in metric measure spaces with irregular boundaries, establishing conditions for the existence and integrability of traces and their implications for zero extension.

Contribution

It introduces new results on the existence and properties of boundary traces of BV functions in irregular metric measure space domains.

Findings

Trace exists in a specific boundary part under certain conditions

The boundary part determines trace integrability and zero extension possibilities

Provides insights into BV functions on irregular metric measure spaces

Abstract

In metric measure spaces, we study boundary traces of BV functions in domains equipped with a doubling measure and supporting a Poincar\'e inequality, but possibly having a very large and irregular boundary. We show that the trace exists in the ordinary sense in a certain part of the boundary, and that this part is sufficient to determine the integrability of the rough trace, as well as the possibility of zero extending the function to the whole space as a BV function.