Local and nonlocal 1-Laplacian in Carnot groups

TL;DR

This paper investigates the local and nonlocal 1-Laplacian problems within Carnot groups, analyzing the transition from nonlocal to local formulations and establishing a key total variation estimate.

Contribution

It introduces a framework for the 1-Laplacian in Carnot groups and proves a total variation estimate crucial for understanding the nonlocal to local limit transition.

Findings

Established a total variation estimate for Carnot groups

Analyzed the passage from nonlocal to local 1-Laplacian problems

Formulated the Dirichlet problem for the 1-Laplace operator in Carnot groups

Abstract

We formulate and study the nonlocal and local least gradient problem, which is the Dirichlet problem for the 1-Laplace operator, in a quite natural setting of Carnot groups. We study the passage from the nonlocal problem to the local problem as the range of the interaction goes to zero; to do this, we first prove a total variation estimate of independent interest.