Quantitative linearization results for the Monge-Ampère equation

Abstract

This paper is about quantitative linearization results for the Monge-Ampère equation with rough data. We develop a large-scale regularity theory and prove that if a measure $\mu$ is close to the Lebesgue measure in Wasserstein distance at all scales, then the displacement of the macroscopic optimal coupling is quantitatively close at all scales to the gradient of the solution of the corresponding Poisson equation. The main ingredient we use is a harmonic approximation result for the optimal transport plan between arbitrary measures. This is used in a Campanato iteration which transfers the information through the scales.

Comment v4: superfluous assumption in Thm 1.2 deleted

References in corpus (2)

• A large-scale regularity theory for the Monge-Ampere equation with rough data and application to the optimal matching problem
• Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space

Cited by in corpus (2)

• Sharp boundary $\epsilon$-regularity of optimal transport maps
• Variational approach to regularity of optimal transport maps: general cost functions