Optimal transportation of vector measures

In this work we study Monge and Kantorovich optimal transportation problems for vector measures. The existence of optimal Kantorovich plans is proved. We formulate the dual problem and prove the equality of optimal values in the primal and dual problem for lower semicontinuous cost functions. The equality of infima in vector-valued Monge and Kantorovich problems is proved under the condition of Lyapunov theorem which guarantees the existence of Monge mappings for vector measures. We give sufficient conditions under which an optimal Kantorovich plan is given by a Monge mapping.