Median values, 1-harmonic functions, and functions of least gradient

TL;DR

This paper explores median value properties of functions, establishing their connection to 1-harmonic functions and proposing a conjecture linking global median properties to least gradient functions.

Contribution

It introduces local and global median value properties for continuous functions and proves their relation to 1-harmonic functions, along with a conjecture about functions of least gradient.

Findings

Functions with local median value property are 1-harmonic in the viscosity sense.

The Dirichlet problem for local median property is either solvable or not, with no middle ground.

A conjecture relates global median property functions to least gradient functions with the same boundary values.

Abstract

Motivated by the mean value property of harmonic functions, we introduce the local and global median value properties for continuous functions of two variables. We show that the Dirichlet problem associated with the local median value property is either easy or impossible to solve, and we prove that continuous functions with this property are 1-harmonic in the viscosity sense. We then close with the following conjecture: a continuous function having the global median value property and prescribed boundary values coincides with the function of least gradient having those same boundary values.