OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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The Kantorovich formulation can be seenas the relaxation of the Monge problem; indeed,if f + is nonatomic we havemin(Kantorovich) = inf(Monge).The existence of an optimal transport mapis a rather delicate question. The first (incomplete)proof is due to Sudakov (1979);various different proofs have been givenby Evans-Gangbo (1999), Trudinger-Wang(2001), Caffarelli-Feldman-McCann (2002),Ambrosio (2003).Wasserstein distance of exponent p: replace( ∫ ) 1/p.the cost byX×X dp (x, y) dγ(x, y)Most of the results remain valid for moregeneral cost functions c(x, y).7
Shape optimization problemsThere is a strong link between mass transportationand shape optimization problems.We present here this relation and we refer tothe papers by Bouchitté-Buttazzo [BB] andBouchitté-Buttazzo-Seppecher [BBS] for alldetails.Shape optimization problem in elasticity:given a force field f in R n find the elasticbody Ω whose “resistance” to f is maximalConstraints:- given volume, |Ω| = m- possible “design region” D given, Ω ⊂ D- possible support region Σ given,Dirichlet regionOptimization criterion: elastic compliance8
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: Monge-Kantorovich problem:{ ∫ }mi
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 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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