Regularity of the Monge-Ampère equation in Besov's space

Calculus of Variations and Partial Differential Equations. 2014. Vol. 49. No. 3-4. P. 1187–1197.

Kolesnikov A., Tikhonov S. Y.

Let \mu = e^{-V} \ dx be a probability measure and T = \nabla \Phi be the optimal transportation mapping pushing forward \mu onto a log-concave compactly supported measure \nu = e^{-W} \ dx. In this paper, we introduce a new approach to the regularity problem for the corresponding Monge--Amp{\`e}re equation e^{-V} = \det D^2 \Phi \cdot e^{-W(\nabla \Phi)} in the Besov spaces W^{\gamma,1}_{loc}. We prove that D^2 \Phi \in W^{\gamma,1}_{loc} provided e^{-V} belongs to a proper Besov class and W is convex. In particular, D^2 \Phi \in L^p_{loc} for some p>1. Our proof does not rely on the previously known regularity results.

Keywords: Besov spaces, пространства Бесова, Optimal transportation, Monge-Ampere equation, a priori estimates, априорные оценки, уравнение Монжа-Ампера, оптимальная транспортировка

Publication based on the results of: Оптимальная транспортировка мер и ее приложения (2012)