The optimal transportation problem was introduced by Monge in 1781. Since then the problem has been extensively studied and more general costs are allowed. But for Monge’s original cost |x-y|, very little is known about the regularity of the optimal mapping. In this talk, we show that, in two dimensional case, the optimal mapping is continuous. By a counter-example we show that the mapping fails to be Lipschitz in general. This is a joint work with F. Santambrogio and X.-J. Wang.

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