Regularity of optimal transport between planar convex domains

TL;DR

This paper establishes global regularity estimates for optimal transport potentials between convex planar domains, advancing understanding of their smoothness and geometric properties.

Contribution

It introduces new techniques for obliqueness and eccentricity growth estimates, leading to $W^{2,p}$-regularity results for transport potentials.

Findings

Proves global $W^{2,p}$-estimates for convex domain transport potentials

Develops new tools for obliqueness in convex domains

Provides bounds on eccentricity growth of potential sections

Abstract

For $0<p<+\infty$, we prove a global $W^{2,p}$-estimate for potentials of optimal transport maps between convex domains in the plane. Among the tools developed for that purpose are obliqueness in general convex domains and estimates for the growth of eccentricity of sections of the potentials.