In these lectures we will describe the relationship between optimal transportation and nonlinear elliptic PDE of Monge-Ampere type, focusing on recent advances in characterizing costs and domains for which the Monge-Kantorovich problem has smooth diffeomorphism solutions.
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R. T . Rachev and L. Ruschendorf, Mass transportation problems, Springer,1998.
J. Urbas, Mass transfer problems, Lecture Notes, Univ. of Bonn, 1998.
C. Villani, Topics in optimal transportation, Amer. Math. Soc. 2003.