### Article

## Римановы метрики в R^n и неравенства типа Соболева

We study the optimal transportation mapping VΦ: ℝd → ℝd pushing forward a probability measure μ = e -V dx onto another probability measure ν = e-W dx. Following a classical approach of E. Calabi we introduce the Riemannian metric g = D2 Φ on ℝd and study spectral properties of the metric-measure space M = (ℝd, g,μ). We prove, in particular, that M admits a non-negative Bakry-Emery tensor provided both V and W are convex. If the target measure ν is the Lebesgue measure on a convex set Ω and μ is log-concave we prove that M is a CD(K, N) space. Applications of these results include some global dimension-free a priori estimates of \\D2 Φ||. With the help of comparison techniques on Riemannian manifolds and probabilistic concentration arguments we proof some diameter estimates for M.

It is well known that Poincarétype inequalities on Riemannian manifolds with measure satisfying the generalized Bakry–Émery condition can be obtained by using the Bochner–Lichnerowicz–Weitzenböck formula. In the case of manifolds with boundary, a suitable generalization is Reilly’s formula. New Poincaré type inequalities for manifolds with measure are obtained by systematically using this formula combined with various conditions on the boundary of the manifold and boundary conditions for elliptic equations. Among other results, a generalization of Colesanti’s inequality, proved earlier in Euclidean space, is presented. It implies a generalization of Brunn–Minkowskitype inequalities for manifolds. A new evolution equation for surfaces on Riemannian manifolds is studied, which determines the Minkowski addition of convex sets in the Euclidean case. The proposed approach covers a large class of convex measures, including measures with heavy tails, which correspond to negative analytic dimension.

It is known that by dualizing the Bochner–Lichnerowicz–Weitzenböck formula, one obtains Poincaré-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry–Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). When the manifold has a boundary, an appropriate generalization of the Reilly formula may be used instead. By systematically dualizing this formula for various combinations of boundary conditions of the domain (convex, mean-convex) and the function (Neumann, Dirichlet), we obtain new Brascamp–Lieb-type inequalities on the manifold. All previously known inequalities of Lichnerowicz, Brascamp–Lieb, Bobkov–Ledoux, and Veysseire are recovered, extended to the Riemannian setting and generalized into a single unified formulation, and their appropriate versions in the presence of a boundary are obtained. Our framework allows to encompass the entire class of Borell’s convex measures, including heavy-tailed measures, and extends the latter class to weighted-manifolds having negative generalized dimension.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.