Partial regularity of brenier solutions of the monge-ampère equation

Abstract

Given Ω,λ C ℝn two bounded open sets, and f and g two probability densities concentrated on and Ω respectively, we investigate the regularity of the optimal map φ (the optimality referring to the Euclidean quadratic cost) sending f onto g. We show that if f and g are both bounded away from zero and infinity, we can find two open sets Ω'C Ω and λ C Ω such that f and g are concentrated on Ω' and λ' respectively, and δ φ':Ω λ is a (bi- Hölder) homeomorphism. This generalizes the 2-dimensional partial regularity result of [8].