Calculus of Variations and Geometric Measure Theory

F. Santambrogio

Variational problems in transport theory with mass concentration

created by santambro on 09 Jan 2007
modified on 28 Nov 2007


Ph.D. Thesis

Inserted: 9 jan 2007
Last Updated: 28 nov 2007

Year: 2006

also published by Edizioni della Normale (2007, Pisa) and distributed by Birkhäuser


The thesis includes the study of several variational problems involving both transport costs and concentration criteria. Some problems are given by the minimization of functional defined on the marginal measures of a transport problem, requiring one to be concentrated, the other diffuse and the two of them close to each other in a transportation sense. This is what is mainly developed in Chapters 1--3. Other chapters are devoted to different models of transportation which involve, in the definition of the cost, sort of concentration criteria about the path particles follow along the transport, giving raise to congestion or branching effects. Several problems are defined on one-dimensional structures (networks, rectifiable supports of vector measures, length constrained sets...) and their regularity, at a blow-up level, is discussed in the last two chapters. The models come in most cases from applications to urban planning, biology (tree, leaves and blood system), geophysics (river basins) and fluid mechanics. the thesis has been written under the supervision of Prof. G. Buttazzo and defended in December at SNS, Pisa.