OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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• if q < 1 + 1/N for every µ 0 and µ 1 thereexists a path giving finite and minimalvalue to the cost J ;• if q ≥ 1 + 1/N there exist measures µ 0(actually a Dirac mass) such that J =+∞ on every nonconstant path startingfrom µ 0 .Open Problems• linking two L q measures in the diffusionunboundedcase;• concentration case in unbounded setting;• Ω unbounded but not necessarily thewhole space;• working with the space W ∞ (Ω);• comparing this model to the ones by Xiaand by Morel, Solimini, . . . ;• numerical computations;• evolution models?63
References(all available at http://cvgmt.sns.it)A. Brancolini, G. Buttazzo: Optimalnetworks for mass transportation problems.Esaim COCV (to appear).G. Buttazzo, A. Davini, I. Fragalà,F. Macià: Optimal Riemannian distancespreventing mass transfer. J. Reine Angew.Math., (to appear).G. Buttazzo, E. Oudet, E. Stepanov:Optimal transportation problems with freeDirichlet regions. In PNLDE 51, BirkhäuserVerlag, Basel (2002), 41–65.G. Buttazzo, A. Pratelli, E. Stepanov:Optimal pricing policies for public transportationnetworks. Preprint Dipartimentodi Matematica Università di Pisa, Pisa (2004).G. Buttazzo, F. Santambrogio: A Modelfor the Optimal Planning of an Urban Area.Preprint Dipartimento di Matematica Universitàdi Pisa, Pisa (2003).64
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OPTIMIZATION PROBLEMS INMASS TRANSP
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The Monge problem is thenmin{ ∫ d
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Then the class of admissible transp
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Monge-Kantorovich problem:{ ∫ }mi
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Shape optimization problemsThere is
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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 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
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OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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