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arXiv · 2607.24666

Global $W^{2,p}$ Regularity in Optimal Transport

Shibing Chen·Yuanyuan Li·Xianduo Wang
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Abstract

In this paper we establish global $W^{2,p}$ estimates for the convex potentials of quadratic optimal transport between bounded convex domains with continuous positive densities. All the assumptions are optimal. The main new ideas include a blow-up analysis that allows for lower-dimensional collapse of the limiting source measure and reduces the limiting problem to a transport problem on its affine hull, and a good-bad scale decomposition in which rigidity controls the good scales while a counting argument shows that the proportion of bad scales tends to zero.

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Shibing Chen, Yuanyuan Li, Xianduo Wang. 2026-07-27. Global $W^{2,p}$ Regularity in Optimal Transport. https://arxiv.org/abs/2607.24666

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