Fine properties of monotone maps arising in optimal transport for non-quadratic costs

TL;DR

This paper investigates the properties of monotone maps in optimal transport with non-quadratic costs, showing they are almost everywhere single-valued and exploring related implications.

Contribution

It establishes that multivalued monotone maps for certain non-quadratic costs are almost everywhere single-valued, extending understanding in optimal transport theory.

Findings

Monotone maps are single-valued almost everywhere for the considered costs.

The study provides new insights into the structure of optimal transport maps with non-quadratic costs.

Consequences for the regularity and structure of solutions are derived.

Abstract

The cost functions considered are $c(x,y)=h(x-y)$, where $h\in C^2(\mathbb{R}^n)$, homogeneous of degree $p\geq 2$, with a positive definite Hessian in the unit sphere. We study multivalued monotone maps with respect to that cost and establish that they are single-valued almost everywhere. Further consequences are then deduced.