Regularity Properties of Optimal Maps Between Nonconvex Domains in the Plane

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Partial regularity for optimal transport maps (2012, background): “In [16, 18] the authors proved the following result:”
The Monge–Ampère equation and its link to optimal transportation (2013, methodology): “However, the advantage of that definition is that it allows for a more refined analysis of the singular set (see [49]).”
On the Monge-Amp\`ere equation (2018, background): “All these facts motivate the following very natural question: What can one say when the convexity assumption on the target is removed? As shown in [33, 25] (see also [29] for a more precise description of the singular set in two dimensions, and [36] for a recent variational proof of the result in [3”
Partial regularity of brenier solutions of the monge-ampère equation (2010, background): “The aim of this paper is to generalize the 2-dimensional result of [8] by showing that there exist two open sets Ω′ ⊂ Ω and Λ′ ⊂ Λ, with |Ω \ Ω′| = |Λ \ Λ′| = 0, such that ϕ is C1,α and strictly convex inside Ω′, and ∇ϕ is a bi-Hölder homeomorphism between Ω′ and Λ′.”