Differentiability of monotone maps related to non-quadratic costs

TL;DR

This paper investigates the differentiability and regularity of monotone maps related to non-quadratic cost functions, establishing local estimates and connecting them to maps of bounded deformation.

Contribution

It introduces new differentiability and regularity results for monotone maps associated with homogeneous costs of degree p ≥ 2, extending previous understanding.

Findings

Local $L^ abla$-estimates of the difference between the map and affine functions

Differentiability of the maps almost everywhere

Hölder continuity properties of the maps

Abstract

The cost functions considered are $c(x,y)=h(x-y)$, with $h\in C^2(R^n)$, homogeneous of degree $p\geq 2$, with positive definite Hessian in the unit sphere. We consider monotone maps $T$ concerning that cost and establish local $L^\infty$-estimates of $T$ minus affine functions, which are applied to obtain differentiability properties of $T$ a.e. It is also shown that these maps are related to maps of bounded deformation, and further, differentiability and H\"older continuity properties are derived.