Partial Regularity of Brenier Solutions of the Monge-ampère Equation

Given Ω,Λ ⊂ R two bounded open sets, and f and g two probability densities concentrated on Ω and Λ 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′ ⊂ Ω and Λ′ ⊂ Λ such that f and g are concentrated on Ω′ and Λ′ respectively, and ∇φ : Ω′ → Λ′ is a (bi-Hölder) homeomorphism. This generalizes the 2-dimensional partial regularity result of [8].