OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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•Large town policy: several ticket pricesWe defined Σ (x, y) = inf{A ( H 1 (Γ \ Σ) )+ B ( H 1 (Γ ∩ Σ) ) : Γ connects x to y}.The cost of the network Σ is defined via theMonge-Kantorovich functional:F (Σ) = MK(f + , f − , d Σ )and the admissible Σ are simply the closedconnected sets with H 1 (Σ) ≤ L.29
Therefore the optimization problem is{}min F (Σ) : Σ cl. conn., H 1 (Σ) ≤ L .Theorem There exists an optimal networkΣ opt for the optimization problem above.In the special case A(t) = t and B ≡ 0(communist model) some necessary conditionsof optimality on Σ opt have been derived(Buttazzo-Oudet-Stepanov 2002 andButtazzo-Stepanov 2003). For instance:• no closed loops;• at most triple point junctions;• 120 ◦ at triple junctions;• no triple junctions for small L;• asymptotic behavior of Σ opt as L → +∞(Mosconi-Tilli JCA 2005);• regularity of Σ opt is an open problem.30
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 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
Page 64 and 65: • if q < 1 + 1/N for every µ 0 a

OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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