OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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∫∫ ∫J(µ) = f(µ a )dm+ f ∞ (µ c )+ g(µ(x))d#ΩΩΩwhere• µ = µ a · m + µ c + µ # is the Lebesgue-Nikodym decomposition of µ with respectto m, into absolutely continuous,Cantor, and atomic parts;• f : R → [0, +∞] is convex, l.s.c.proper;• f ∞ is the recession function of f;and• g : R → [0, +∞] is l.s.c. and subadditive,with g(0) = 0;• # is the counting measure;• f and g verify the compatibility conditionlimt→+∞f(ts)t= limt→0 + g(ts)t.55
In this way the functional J is l.s.c. for theweak* convergence of measures. If we furtherassume that f(s) > 0 for s > 0 andg(1) > 0, then we have J ≥ c > 0. The existencetheorem then applies and, given twoprobabilities µ 0 and µ 1 , we obtain at leasta minimizing path for the problem}min{J (γ) : γ(0) = µ 0 , γ(1) = µ 1 .provided J is not identically +∞.We now study two special cases (Ω ⊂ R Ncompact and m = dx):Concentration f ≡ +∞, g(z) = |z| rr ∈]0, 1[. We have then∫J(µ) = |µ(x)| r d#Ωµ atomicwithDiffusion f(z) = |z| q with q > 1, g ≡ +∞.We have then∫J(µ) = |u(x)| q dx µ = u · dx, u ∈ L q .Ω56
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OPTIMIZATION PROBLEMS INMASS TRANSP
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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 52 and 53: We also give a model where the oppo
Page 54 and 55: Theorem Assume:• J is lower semic
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
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OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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