IUMJ

Title: Functions of least gradient and 1-harmonic functions

Authors: Jose M. Mazon, Julio Rossi and Sergio S. de Leon

Issue: Volume 63 (2014), Issue 4, 1067-1084

Abstract:

In this paper, we find the Euler-Lagrange equation corresponding to functions of least gradient. It turns out that this equation can be identified with the $1$-Laplacian. Moreover, given a Lipschitz domain $\Omega$, we prove that there exists a function of least gradient in $\Omega$ that extends every datum belonging to $L^1(\partial\Omega)$. We show, as well, the non-uniqueness of solutions in the case of discontinuous boundary values.