Functions of least gradient and 1-harmonic functions J. M. MazonJulio RossiS. Segura de Leon 35J7535J2035J9235J25Functions of least gradient1-Laplacian 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. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5327 10.1512/iumj.2014.63.5327 en Indiana Univ. Math. J. 63 (2014) 1067 - 1084 state-of-the-art mathematics http://iumj.org/access/