Abstract

We prove the existence of solutions to the Monge problem with an absolutely continuous initial measure by solving a secondary variational problem with any strictly convex function, the so-called secondary variational method. The cost function is given by an arbitrary norm on ${ℝ}^n$. In addition, if a norm satisfies the uniform smoothness and convexity estimates, and two measures are absolutely continuous, then for the Monge problem with such a norm cost function, we can find a same optimal transport map via the secondary variational method even with different strictly convex functions (the classical Monge problem is a special case). This optimal transport map is just the one which uniquely satisfies a monotone condition. Finally, we construct an example with the $L^1$ norm cost function, which is not a strictly convex norm, to show that one can get different optimal transport maps by solving secondary variational problems with different strictly convex functions. In view of this example, for the Monge problem between two absolutely continuous measures, if a norm cost function does not satisfy the uniform smoothness and convexity estimates, there can be no uniqueness of optimal transport maps obtained via the secondary variational method.

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