Stability and continuity of functions of least gradient

TL;DR

This paper proves that functions of least gradient on metric measure spaces are continuous outside their jump sets and introduces stability properties, with applications to maximum principles in Dirichlet problems.

Contribution

It establishes continuity properties and stability results for least gradient functions on metric measure spaces, extending understanding of their regularity and boundary behavior.

Findings

Functions of least gradient are continuous outside jump sets.

Develops stability properties for sequences of least gradient functions.

Proves a maximum principle for solutions to Dirichlet problems.

Abstract

In this note we prove that on metric measure spaces, functions of least gradient, as well as local minimizers of the area functional (after modification on a set of measure zero) are continuous everywhere outside their jump sets. As a tool, we develop some stability properties of sequences of least gradient functions. We also apply these tools to prove a maximum principle for functions of least gradient that arise as solutions to a Dirichlet problem.