OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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and so the value of the cost F (a). The classof admissible controls is taken as{A = a(x) Borel measurable :∫}α ≤ a(x) ≤ β, a(x) dx ≤ m .In the case when f + = δ x and f − = δ yare Dirac masses concentrated on two fixedpoints x, y ∈ Ω, the problem of maximizingF (a) is nothing else than that of provingthe existence of a conformally flat Euclideanmetric whose geodesics joining x and y areas long as possible.This problem has several natural motivations;indeed, in many concrete examples,one can be interested in making as difficultas possible the communication betweenΩsome masses f + and f − .For instance, itis easy to imagine that this situation may41
arise in economics, or in medicine, or simplyin traffic planning, each time the connectionbetween two “enemies” is undesired.Of course, the problem is made non trivialby the integral constraint ∫ a(x) dx ≤ m,Ωwhich has a physical meaning: it prescribesthe quantity of material at one’s disposal tosolve the problem.The analogous problem of minimizing thecost functional F (a) over the class A, whichcorresponds to favor the transportation off + into f − , is trivial, sinceinf{}F (a) : a ∈ A= F (α).The existence of a solution for the maximizationproblemmax{}F (a) : a ∈ A42
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 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 52 and 53: We also give a model where the oppo
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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