Equivalent formulations for Monge–Kantorovich equation

In this paper, we study some equivalent formulations in divergence form for the optimization problem max{∫Ωξdμ;ξ∈W01,1(Ω) s.t. |∇ξ(x)|≤k(x) a.e. x∈Ω} where k∈C(Ω¯) and k>0k>0 in ΩΩ. This is the so called dual equation of Monge–Kantorovich problem.