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Regularity Properties of Optimal Maps Between Nonconvex Domains in the Plane

Given two bounded open subsets Ω, Λ ⊂ ℝ d , and two densities f and g concentrated on Ω and Λ, respectively, we investigate the regularity of the optimal map ∇ϕ sending f onto g. We show that, if f and g are both bounded away from zero and infinity, then 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 homeomorphism. Moreover, if f and g are smooth, then ∇ϕ is a smooth diffeomorphism between Ω′ and Λ′. Finally, we give a quite precise description of the singular set of ϕ, showing that it is a 1-dimensional manifold of class C 1 out of a countable set.