OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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produces an optimal µ which has a onedimensionalconcentration in the vertical segment.Then a theory of variational problems withrespect to measures has to be developed.This has been done by Bouchitté, Buttazzoand Seppecher [Calc. Var. 1997]. The applicationof transport problems to mass optimizationhas been developed by Bouchittéand Buttazzo [JEMS 2001]. The key toolsare:• the tangent space T µ (x) to a measure µ,defined for µ a.e. x;• the tangential gradient D µ ;• the Sobolev spaces W 1,pµ .19
Then the transport problem becomes equivalentto the PDE⎧⎨ − div ( µ(x)D µ u ) = f in R n \ Σu is 1-Lipschitz on D, u = 0 on Σ⎩|D µ u| = 1 µ-a.e. on R n , µ(Σ) = 0.Theorem. For every measure f thereexists a solution µ opt of the mass optimizationproblem. Moreover, a measure µsolves the mass optimization problem if andonly if (a multiple of) it solves the Monge-Kantorovich PDE.In order to describe the tangential gradientsD µ u which appear in the Monge-Kantorovich equation above we recall themain steps of the theory of the variationalintegrals w.r.t. a measure.20
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 18 and 19: More precisely, we consider the geo
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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