OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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Theorem Let (X, d, d ′ ) be a metric spacewith two different distances. Assume that:• d ′ ≤ d;• all d-bounded sets in X are relativelycompact with respect to d ′ ;• the mapping d : X × X → R + is a lowersemicontinuous function with respect tothe distance d ′ × d ′ .Consider a functional J : X → [0, +∞] andassume that:• J is d ′ -l.s.c.;• ∫ +∞ (infBd0(r) J ) dr = +∞.Then the functionalJ (γ) =∫ 10J(γ(t))|γ ′ | d (t) dthas a minimizer on the set of d−Lipschitzcurves connecting two given points x 0 andx 1 , provided it is not identically +∞.61
As an application of the theorem above weconsider the diffusion case when Ω = R N .The concentration case in the unboundedsetting still presents some extra difficultiesthat we did not yet solve. We take:• d the Wasserstein distance;• d ′ the distance metrizing the weak* topologyon probabilities:d ′ (µ, ν) =∞∑k=12 −k1 + c k∣ ∣〈φk , µ − ν〉 ∣ ∣where (φ k ) is a dense sequence of Lipschitzfunctions in the unit ball of C b (Ω)and c k are their Lipschitz constants.We may prove that the assumptions of theabstract scheme are fulfilled, and we have(for the diffusion case):62
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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
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 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 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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