The Euler-Lagrange equation for the Anisotropic least gradient problem

In this paper we find the Euler-Lagrange equation for the anisotropic least gradient problem inf{∫Ωϕ(x,Du):u∈BV(Ω),u|∂Ω=f} being ϕ a metric integrand and f∈L1(∂Ω). We also characterize the functions of ϕ-least gradient as those whose boundary of the level set is ϕ-area minimizing in Ω.