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Regularity of Potential Functions of the Optimal Transportation Problem

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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.

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Correspondence to Neil S. Trudinger.

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Communicated by L. Ambrosio

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Ma, XN., Trudinger, N. & Wang, XJ. Regularity of Potential Functions of the Optimal Transportation Problem. Arch. Rational Mech. Anal. 177, 151–183 (2005).

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