OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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More precisely, we consider the geodesic distanceon D × D{ ∫ 1d D (x, y) = min |γ ′ (t)| dt :0}γ(t) ∈ D, γ(0) = x, γ(1) = y= sup { |ϕ(x) − ϕ(y)| :|Dϕ| ≤ 1 on D }and its modification by Σ{d D,Σ (x, y) = inf d D (x, y) ∧ ( d D (x, ξ 1 )+ d D (y, ξ 2 ) ) }: ξ 1 , ξ 2 ∈ Σ= sup { |ϕ(x) − ϕ(y)| :|Dϕ| ≤ 1 on Ω, ϕ = 0 on Σ } .This semi-distance will play the role of costfunction in the Monge-Kantorovich masstransport problem.17
When f = f + − f − is a smooth function,D = R n , Σ = ∅, Evans and Gangbo [Mem.AMS 1999] have shown that the solution canbe found by solving the PDE⎧⎨ − div ( a(x)Du ) = fu is 1-Lipschitz⎩|Du| = 1 a.e. on {a(x) > 0}where the coefficient a is a multiple of theoptimal µ.However, even simple cases with f ∈ H −1may produce optimal mass distributionswhich are singular measures. For instancef +f - 18
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 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
Page 64 and 65: • if q < 1 + 1/N for every µ 0 a

OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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