p-harmonic approximation of functions of least gradient

The purpose of this note is to establish a natural connection between the minimizers of two closely related variational problems. We prove global and local convergence results for the p-harmonic functions, defined as continuous local minimizers of the LP norm of the gradient for 1 < p < infinity, as p -> 1, and show that the limit function minimizes at least locally the total variation of the vector-valued measure del u in BV(Omega).