Article

Archive for Rational Mechanics and Analysis

, Volume 177, Issue 2, pp 151-183

Regularity of Potential Functions of the Optimal Transportation Problem

  • Xi-Nan MaAffiliated withCentre for Mathematics and Its Applications, The Australian National University
  • , Neil S. TrudingerAffiliated withCentre for Mathematics and Its Applications, The Australian National University Email author 
  • , Xu-Jia WangAffiliated withCentre for Mathematics and Its Applications, The Australian National University

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Abstract

The potential function of the optimal transportation problem satisfies a partial differential equation of Monge-Ampère type. In this paper we introduce the notion of a generalized solution, and prove the existence and uniqueness of generalized solutions of the problem. We also prove the solution is smooth under certain structural conditions on the cost function.