Least gradient functions in metric random walk spaces

TL;DR

This paper investigates least gradient functions within metric random walk spaces, including graphs and nonlocal Euclidean spaces, establishing conditions for their Euler-Lagrange equations and proving related Poincaré inequalities.

Contribution

It introduces a unified framework for least gradient functions in metric random walk spaces and derives new results on their Euler-Lagrange equations and Poincaré inequalities.

Findings

Euler-Lagrange equations for least gradient functions derived

Poincaré inequality established in various settings

Framework unifies local and nonlocal least gradient problems

Abstract

In this paper we study least gradient functions in metric random walk spaces, which include as particular cases the least gradient functions on locally finite weighted connected graphs and nonlocal least gradient functions on $\mathbb{R}^N$. Assuming that a Poincar\'e inequality is satisfied, we study the Euler-Lagrange equation associated with the least gradient problem. We also prove the Poincar\'e inequality in a few settings.