Chen, ShibingFigalli, Alessio2015-06-012015-06-010001-8708http://hdl.handle.net/1885/13661We develop an ε-regularity theory at the boundary for a general class of Monge–Ampère type equations arising in optimal transportation. As a corollary we deduce that optimal transport maps between Hölder densities supported on C2 uniformly convex domains are C1,α up to the boundary, provided that the cost function is a sufficient small perturbation of the quadratic cost −x · y.The second author has been partially supported by NSF Grant DMS-1262411. This material is based upon work supported by the National Science Foundation under Grant No. 0932078 000.28 pages© 2015 Elsevier Inc. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ http://www.sherpa.ac.uk/romeo/issn/0001-8708/ Author can archive post-print (ie final draft post-refereeing) after 24 months embargo (Sherpa/Romeo 30/7/2017)Optimal transportationMonge–AmpèreBoundary regularityGeneral cost201510.1016/j.aim.2014.12.0322016-02-24