OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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The shape optimization problem above hasin general no solution; in fact minimizing sequencesmay develop wild oscillations whichgive raise to limit configurations that are notin a form of a domain.Therefore, in order to describe the behaviourof minimizing sequences a relaxedformulation is needed.The literature on this problem is very wide;starting from the works by Murat and Tartarin the ’70, it has been studied bymany authors (Kohn and Strang, Lurie andCherkaev, Allaire, Bendsøe, . . . ). There arestrong links with homogenization theory.A large parallel literature is present in engineering.11
The complete relaxed form of the problemabove is not known: several difficulties arise.For instance:• Lack of coercivity; in most of the literaturemixtures of two homogeneous isotropicmaterials are considered. Hereone of the two materials is the emptyspace (outside Ω).• Even for mixtures of two materials in theelasticity case the set of materials attainableby relaxation is not known.• Possibility of nonlocal problems as limitsof admissible sequences of domains.• When D is unbounded, it is not clearthat data with compact support provideminimizing sequences of domains thatare equi-bounded.12
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 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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