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Regularity of potential functions of the optimal transportation problem. (English) Zbl 1072.49035

Summary: 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.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
35A15 Variational methods applied to PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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