Remarks on the stability of mass minimizing currents in the Monge-Kantorovich problem

The Monge-Kantorovich problem is equivalent to the problem of finding 1-currents with fixed boundary and minimal mass. We address the question of the stability for the mass minimizing currents. In particular, we state a Γ-convergence result. We provide proofs relying just on basic properties of currents and on the notion of flat norm.