Quantitative linearization results for the Monge-Ampère equation

Michael Goldman (1) , Martin Huesmann (2) , Felix Otto (3)

1 LJLL - Laboratoire Jacques-Louis Lions
2 WWU - Westfälische Wilhelms-Universität Münster = University of Münster
3 MPI-MiS - Max Planck Institute for Mathematics in the Sciences

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 µ 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.

Domaines: Equations aux dérivées partielles [math.AP]

Michael Goldman, Martin Huesmann, Felix Otto. Quantitative linearization results for the Monge-Ampère equation. Communications on Pure and Applied Mathematics, In press. ⟨hal-02137730v4⟩