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The book mainly deals with variational problems involving optimal transportation of probability measures, in competition with concentration effects. First, some problems are presented where measures have to be selected minimizing transport costs between them but satisfying some concentration criteria, with possible applications mainly to urban planning (where the concentrated measures stand for services in the city and the diffuse ones for population). The second part of the book is devoted to optimization problems where the concentration or diffusion phenomena occurs directly at the level of the transportation structure: for instance in most communication networks, as well as in river basins and blood vessels joint transportation is favoured, while some models for traffic congestion or compressible fluid mechanics give raise to problems where spread configurations are preferred.
List of contents
Preliminaries on optimal transportation.- 1. An urban planning model by local functionals.- 2. An urban planning model with traffic congestion.- 3. Transport and concentration problems with interaction.- 4. Path functionals in Wasserstein spaces.- 5. A system of PDEs from a geodesic problem in Wp.- 6. Branching transport problems and distances.- 7. Landscape function.- 8. Blow-up for optimal one-dimensional sets.- 9. Blow-up for optimal branching structures.- References.
Report
From the reviews: "The book deals with optimal transport problems where concentration phenomena, such as concentration of the marginal measures or along the transport itself, occur. The content of the volume is divided into 10 chapters preceded by an Introduction outlining the main results obtained by the author. ... the text will be of real interest for applied mathematicians." (Adriana-Ioana Lefter, Memoirs of the Scientific Sections, 2009)
