Abstract

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; Plusieurs problèmes d'optimisation liés à la théorie du transport optimal, concernant aussi des critères de concentration, sont étudiés. Il s'agit, pour ce qui est des primiers chapitres, de la minimization de fonctionnelles definies sur les mesures marginales du porblème de transport, en demandant que l'une soit concentrée et l'autre diffusée, alors que les deux doivent être proche au sense du transport de masse. D'autres chapitres portent sur des modèles différentes qui considèrent la concentration des parcours suivis par les particules lors du mouvement, en donnant des effets de congestion ou branchement. Plusieurs problèmes font apparaître des structures de dimension 1 (reseaux, supports rectifiables de mesures vectorielles, ensemble sous contraintes de longueur...) et leur régularité (blow-up) est étudiée dans les deux derniers chapitres. Les modèles viennent dans la majorité des cas de possibles applications à la planification urbaine, la biologie (arbres, feuilles et système sanguin), la géophysique (bassins fluviaux) et la mécanique des fluides. La thèse a été écrite sous la direction du Prof. Buttazzo et soutenue à l'Ecole Normale Supérieure de Pise.

Abstract

Congested traffic problems on very dense networks lead, at the limit, to minimization problems posed on measures on curves as shown in Baillon and Carlier (Netw. Heterog. Media 7:219–241, 2012 ). Here, we go one step further by showing that these problems can be reformulated in terms of the minimization of an integral functional over a set of vector fields with prescribed divergence as in Beckmann (Econometrica 20:643–660, 1952 ). We prove a Sobolev regularity result for their minimizers despite the fact that the Euler–Lagrange equation of the dual is highly degenerate and anisotropic. This somehow extends the analysis of Brasco et al. (J. Math. Pures Appl. 93:652–671, 2010 ) to the anisotropic case. Copyright Springer Science+Business Media New York 2013