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Optimality conditions and exact solutions of the two-dimensional Monge-Kantorovich problem. (English. Russian original) Zbl 1099.49028
J. Math. Sci., New York 133, No. 4, 1456-1463 (2006); translation from Zap. Nauchn. Semin. POMI 312, 150-164, 314 (2004).
Let $$X, Y$$ be two Polish spaces, $$\sigma$$ (resp. $$\tau)$$ a positive Borel measure in $$X$$ (resp. in $$Y)$$ with $$\sigma(X) = \tau(Y),$$ $$c : X \times Y \to {\mathbb R}$$ a bounded continuous function (the cost function). The Monge-Kantorovich problem (1941) consists of minimizing the functional $\langle c, \mu \rangle = \int_{X \times Y} c(x, y) \mu(dx \,dy)$ (the cost of mass transfer) in the set of all positive Borel measures $$\mu$$ in $$X \times Y$$ having $$\sigma, \tau$$ as marginals, which means $$\mu({\mathcal B} \times Y) = \sigma({\mathcal B})$$ for every Borel set $$\mathcal B$$ in $$X$$ and $$\mu(X \times {\mathcal C}) = \tau({\mathcal C})$$ for every Borel set $$\mathcal C$$ in $$Y.$$ This problem is a relaxation of the Monge problem (1781) where (in modern formulation) the functional to be minimized is $F(f) = \int_X c(x, f(x))\sigma(dx)$ over all measure preserving Borel maps $$f : (X, \sigma) \to (Y, \tau)$$ (this means $$f(\sigma) = \tau,$$ or, more precisely, $$\tau({\mathcal C}) = \sigma(f^{-1}({\mathcal C}))$$ for every Borel set $$\mathcal C$$ in $$Y).$$ The author gives optimality conditions for both problems and explicit solutions to various two dimensional Monge and Monge-Kantorovich problems. Not much prior knowledge is assumed from the reader, numerous references are given and the emphasis is on explicit solutions, thus this paper is a good introduction to Monge and Monge-Kantorovich problems. For a paper with somewhat related aims but different contents see A. Caffarelli [Lect. Notes Math. 1813, 1–10 (2003; Zbl 1065.49027)].

##### MSC:
 49Q20 Variational problems in a geometric measure-theoretic setting 49J45 Methods involving semicontinuity and convergence; relaxation
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