Global $W^{2,1+ε}$ regularity for potentials of optimal transport of non-convex planar domains

TL;DR

This work addresses global regularity for Alexandrov solutions of the Monge-Ampère equation arising in optimal transport when the source domain $\Omega$ is a bounded non-convex polygon in $\mathbb{R}^2$ and the target domain $\Omega^*$ is convex, with densities $f,g$ bounded away from $0$ and infinity. The authors develop localisation and a doubling property for small-height sections of the potential $u$, and they carefully analyze geometry near concave vertices by decomposing the domain and employing engulfing and Vitali-covering ideas. The main result is a global $W^{2,1+\epsilon}$ estimate for $u$, with constants depending only on $\lambda$ and the geometry of the domains, extending previous two-dimensional results to polygonal non-convex sources. The findings have implications for numerical methods and mesh generation on polygonal domains where optimal transport-based mass redistribution is used.

Abstract

In this paper, we investigate the optimal transport problem when the source is a non-convex polygonal domain in $\mathbb{R}^2$. We show a global $W^{2,1+ε}$ estimate for potentials of optimal transport. Our method applies to a more general class of domains.

Figures (1)

• Figure 3.1:

Theorems & Definitions (24)

• Theorem 1.1
• Proposition 1.1
• Definition 2.1
• Definition 2.2
• Remark 2.1
• Lemma 2.1: John's lemma J
• Lemma 3.1
• proof
• Lemma 3.2
• proof