On nonlinear Kantorovich problems for cost functions of a special form

TL;DR

This paper investigates nonlinear optimal transportation problems with special cost functions, establishing existence results and linking them to Monge problems with convex dominance, advancing understanding of complex transportation models.

Contribution

It introduces a class of nonlinear Kantorovich problems with specific cost functions and proves existence or non-existence of solutions, connecting to Monge problems with convex dominance.

Findings

Existence of solutions for certain nonlinear Kantorovich problems

Non-existence results under specific conditions

Connection established between nonlinear Kantorovich and Monge problems

Abstract

We study Kantorovich type optimal transportation problems with nonlinear cost functions, including dependence on conditional measures of transport plans. A range of nonlinear Kantorovich problems for cost functions of a special form is considered and results on existence (or non-existence) of optimal solutions are proved. We also establish the connection between the nonlinear Kantorovich problem with the cost function of some special form and the Monge problem with convex dominance.