Local and nonlocal 1-Laplacian 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 the non-Euclidean 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. During this procedure, we prove a total variation estimate of independent interest and give an existence result for the local problem.