OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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• there is a transportation cost for movingfrom the residential areas to the servicespoles;• people desire not to live in areas wherethe density of population is too high;• services need to be concentrated as muchas possible, in order to increase efficiencyand decrease management costs.The transportation cost will be describedthrough a Monge-Kantorovich mass transportationmodel; it is indeed given by a p-Wasserstein distance (p ≥ 1) W p (f + , f − ),being p = 1 the classical Monge case.The total unhappiness of citizens due tohigh density of population will be describedby a penalization functional, of the form{ ∫H(f + h(u) dx) = Ω if f + = u dx+∞ otherwise,37
where h is assumed convex and superlinear(i.e. h(t)/t → +∞ as t → +∞). The increasingand diverging function h(t)/t thenrepresents the unhappiness to live in an areawith population density t.Finally, there is a third term G(f − ) whichpenalizes sparse services. We force f − tobe a sum of Dirac masses and we considerG(f − ) a functional defined on measures,of the form studied by Bouchitté-Buttazzo(Nonlinear An. 1990, IHP 1992, IHP 1993):{ ∑G(f − ) = n g(a n) if f − = ∑ n a nδ xn+∞ otherwise,where g is concave and with infinite slopeat the origin. Every single term g(a n ) inthe sum represents here the cost for buildingand managing a service pole of dimensiona n , located at the point x n ∈ Ω.38
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 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 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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