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Nonlinear mobility continuity equations and generalized displacement convexity arXiv:0901.3978v1 [math.AP
"... We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the intern ..."
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Cited by 5 (2 self)
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We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As byproduct, we obtain existence, stability, and contraction results for the semigroup obtained by solving the homogeneous Neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain. For the potential energy and the interaction energy, we present a nonrigorous argument indicating that they are not displacement semiconvex.
AN EXISTENCE RESULT FOR THE MONGE PROBLEM IN R^n WITH NORM COST FUNCTIONS
, 2009
"... We establish existence of solutions to the Monge problem in n with a norm cost function, assuming absolute continuity of the initial measure. The loss in strict convexity of the unit ball implies that transport is possible along several directions. As in [4], we single out particular solutions to t ..."
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Cited by 3 (0 self)
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We establish existence of solutions to the Monge problem in n with a norm cost function, assuming absolute continuity of the initial measure. The loss in strict convexity of the unit ball implies that transport is possible along several directions. As in [4], we single out particular solutions to the Kantorovich relaxation with a secondary variational problem, which involves a strictly convex norm. We then define a map rearranging the mass within the rays, by a Sudakovtype argument with the disintegration technique in [8, 10, 17]. In the secondary variational problem the cost is also infinite valued and in general there is no Kantorovich potential. However, all the optimal transport plans share the same maximal transport rays. We derive also an expression for the transport density associated to these optimal plans. Remark: The construction presently given in the preprint needs the further technical assumption that, with the notation of Section 3.4, the set of points x ∈ Ts whose secondary transport ray belongs to the
Devant le jury composé de:
"... sous la coordination de Ivar Ekeland Problèmes classiques et moins classiques en transport optimal Régularité, approximation, EDP et applications ..."
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sous la coordination de Ivar Ekeland Problèmes classiques et moins classiques en transport optimal Régularité, approximation, EDP et applications
Models and applications of Optimal Transport in Economics, Traffic and Urban Planning
, 2009
"... 1.1 Generalizations of Beckmann’s Problem......................... 2 1.2 Wardrop equilibria, the discrete case........................... 4 ..."
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1.1 Generalizations of Beckmann’s Problem......................... 2 1.2 Wardrop equilibria, the discrete case........................... 4
Models and applications of Optimal Transport Theory
, 2009
"... These lecture notes will present the main issues and ideas of some variational problems that use or touch the theory of Optimal Transportation. Just ideas, almost no proofs. 1 The urban planning of residents and services A very simplified model that has been proposed for studying the distribution of ..."
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These lecture notes will present the main issues and ideas of some variational problems that use or touch the theory of Optimal Transportation. Just ideas, almost no proofs. 1 The urban planning of residents and services A very simplified model that has been proposed for studying the distribution of residents and services in a given urban region Ω passes through the minimization of a total quantity F(µ, ν) concerning two unknown densities µ and ν. the two measures µ and ν will be searched among probabilities on Ω. This means that the total amounts of population and production are fixed as problem data. The definition of the total cost functional to optimize takes into account some criteria we want the two densities µ and ν to satisfy: (i) there is a transportation cost T for moving from the residential areas to the services areas; (ii) people do not want to live in areas where the density of population is too high; (iii) services need to be concentrated as much as possible in order to increase efficiency and decrease management costs. Fact (i) is described, in its easiest version, through a pWasserstein distance (p ≥ 1). We will look at T (µ, ν) = W p p (µ, ν). Fact (ii) will be described by a penalization functional, a kind of total unhappiness of citizens due to high density of population, obtained by integrating with respect to the citizens ’ density their personal unhappiness. Fact (iii) is modeled by a third term representing costs for managing services once they are located according to the distribution ν, taking into account that efficiency depends strongly on how much ν is concentrated. The cost functional to be considered is then
A DACOROGNAMOSER APPROACH TO FLOW DECOMPOSITION AND MINIMAL FLOW PROBLEMS
"... Abstract. The papers describes an easy approach, based on a classical construction by Dacorogna and Moser, to prove that optimal vector fields in some minimal flow problem linked to optimal transport models (congested traffic, branched transport, Beckmann’s problem...) are induced by a probability m ..."
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Abstract. The papers describes an easy approach, based on a classical construction by Dacorogna and Moser, to prove that optimal vector fields in some minimal flow problem linked to optimal transport models (congested traffic, branched transport, Beckmann’s problem...) are induced by a probability measure on the space of paths. This gives a new, easier, proof of a classical result by Smirnov, and allows handling optimal flows without taking care of the presence of cycles. hal00871623, version 1 10 Oct 2013