Partial regularity of brenier solutions of the monge-ampère equation

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The Monge–Ampère equation and its link to optimal transportation (2013, methodology): “In the quadratic cost case, in [49, 51] it was shown that on a big set there is still good control on the Monge–Ampère measure of the potential u allowing us to apply the local theory of classical Monge–Ampère equation; see the sketch of the proof of Theorem 3.”
Partial regularity for optimal transport maps (2012, background): “In [16, 18] the authors proved the following result: Theorem 1.2.”