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Lecture notes on optimal transport problems. (English) Zbl 1047.35001
Colli, Pierluigi (ed.) et al., Mathematical aspects of evolving interfaces. Lectures given at the C.I.M.-C.I.M.E. joint Euro-summer school, Madeira, Funchal, Portugal, July 3–9, 2000. Berlin: Springer (ISBN 3-540-14033-6/pbk). Lect. Notes Math. 1812, 1-52 (2003).
These 52 pages long lecture notes deals with the Monge-Kantorovich optimal transport problem. The original formulation of Monge can be stated as “given two distributions of equal masses of given two material functions \(g^{\pm}(x)\), find a transport map \(\xi\) which carries the first distribution into the second one and minimizes the transport cost \(C(\xi)\).”
The lecture notes are divided into 9 sections. The first section contains some elementary examples. The second section discusses existence and regularity of optimal transport plans. The third and fourth sections are dedicated to the one-dimensional case and the ODE version of the optimal transport problem. The fifth section deals with the PDE version of the optimal transport problem and the \(p\)-Laplacian approximation. Existence of optimal transport maps are studied in section six, while the regularity and uniqueness of the transport density is discussed in section seven. The eighth section is dedicated to the study of the Bouchitté-Buttazzo mass optimization problem. The last, ninth section, is an appendix which summarizes some necessary measure theoretic results.
For the entire collection see [Zbl 1013.00023].

49Q20 Variational problems in a geometric measure-theoretic setting
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
49N60 Regularity of solutions in optimal control
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B37 PDE in connection with control problems (MSC2000)