Interior C 2,α Regularity for Potential Functions in Optimal Transportation

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The Monge–Ampère equation and its link to optimal transportation (2013, background): “3 to this general setting have been obtained in [84, 85, 38].”
Hölder regularity of optimal mappings in optimal transportation (2009, methodology): “From this property we obtain the C 1,α regularity of the solution u. The uniform boundednessof A underlineartransformationsisalsocrucialforthe C 2,α regularityofthepotential functions, treated in a subsequent paper [ 16 ], which extends the C 2,α estimate in [13].”
Five lectures on optimal transportation: Geometry, regularity and applications (2010, background): “More generally, we claim u inherits Lipschitz and semiconvexity bounds (23)–(24) from c(x, y), which guarantee the existence of x-derivatives in the preceding corollary — at least Lebesgue almost everywhere.”
Hölder Continuity and Injectivity of Optimal Maps (2011, background): “Necessity of Trudinger and Wang’s condition for continuity was shown by Loeper [42], who noted its covariance (as did [35] [49]) and some relations to curvature.”