OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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We also give a model where the oppositefeature occurs: instead of favouring the concentrationof transport rays, the variationalfunctional prevents it giving a lower cost todiffused measures.The mathematical model will be consideredin an abstract metric space frameworkwhere both behaviours (favouring concentrationand diffusion) are included; we donot know of natural phenomena where thediffusive behaviour occurs.The results presented here are contained inA. Brancolini, G. Buttazzo, F. Santambrogio:Path functionals over Wassersteinspaces. Preprint 2004, available athttp://cvgmt.sns.it51
We consider an abstract framework wherea metric space X with distance d is given.We assume that closed bounded subsets ofX are compact. Fix two points x 0 and x 1in X and consider the path functionalJ (γ) =∫ 10J(γ(t))|γ ′ |(t) dtwhere γ : [0, 1] → X ranges among allLipschitz curves such that γ(0) = x 0 andγ(1) = x 1 . Here:• J : X → [0, +∞] is a given mapping onX;• |γ ′ |(t) is the metric derivative of γ at thepoint t, i.e.|γ ′ |(t) = lim sups→td ( γ(s), γ(t) )|s − t|.52
Page 1 and 2: OPTIMIZATION PROBLEMS INMASS TRANSP
Page 3 and 4: The Monge problem is thenmin{ ∫ d
Page 5 and 6: Then the class of admissible transp
Page 7 and 8: Monge-Kantorovich problem:{ ∫ }mi
Page 9 and 10: Shape optimization problemsThere is
Page 12 and 13: The shape optimization problem abov
Page 14 and 15: To overcome some of these difficult
Page 16 and 17: In the mass optimization problem th
Page 18 and 19: More precisely, we consider the geo
Page 20 and 21: produces an optimal µ which has a
Page 22 and 23: Variational integrals w.r.t. a meas
Page 24 and 25: Here are some cases where the optim
Page 26 and 27: -2 -1.5 -1 -0.5 0 0.5 1 1.5 21.510.
Page 28 and 29: Problem 1 - Optimal NetworksWe cons
Page 30 and 31: •Large town policy: several ticke
Page 32 and 33: It is interesting to study the opti
Page 34 and 35: The function B can be seen as a con
Page 36 and 37: Here is another case, with a single
Page 38 and 39: • there is a transportation cost
Page 40 and 41: We have then the optimization probl
Page 42 and 43: and so the value of the cost F (a).
Page 44 and 45: is a delicate matter. Indeed, maxim
Page 46 and 47: Some Numerical ComputationsHere are
Page 48 and 49: Optimal sets of length 0.5 and 1 in
Page 50 and 51: Now we give a model that takes into
Page 54 and 55: Theorem Assume:• J is lower semic
Page 56 and 57: ∫∫ ∫J(µ) = f(µ a )dm+ f ∞
Page 58 and 59: Concentration case. The following f
Page 60 and 61: Diffusion case. The following facts
Page 62 and 63: Theorem Let (X, d, d ′ ) be a met
Page 64 and 65: • if q < 1 + 1/N for every µ 0 a

OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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