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A Monge-Kantorovich mass transport problem for a discrete distance. (English) Zbl 1225.49047

Summary: This paper is concerned with a Monge-Kantorovich mass transport problem in which in the transport cost we replace the Euclidean distance with a discrete distance. We fix the length of a step and the distance that measures the cost of the transport depends of the number of steps that is needed to transport the involved mass from its origin to its destination. For this problem we construct special Kantorovich potentials, and optimal transport plans via a nonlocal version of the PDE formulation given by Evans and Gangbo for the classical case with the Euclidean distance. We also study how these problems, when rescaling the step distance, approximate the classical problem. In particular we obtain, taking limits in the rescaled nonlocal formulation, the PDE formulation given by Evans-Gangbo for the classical problem.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
49J21 Existence theories for optimal control problems involving relations other than differential equations
90B06 Transportation, logistics and supply chain management
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