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8 Optimal Transportation and Monge–Ampère Equations
132

where the infimum is taken over all transportation maps S, which are one-to-one
onfromX toY,measurableandpushthemeasuref intog.Obviously,thisisavery
difficult optimisation problem, mainly due to the highly nonlinear constraint
(8.1) on S and due to the seemingly total lack of compactness of minimizing
sequences. No derivative of S is involved in C c , which might give coercivity!
A big step forward was taken by L.V. Kantorovich in the ’40 s (see [11], [12]).
He introduced the following relaxed version of the Monge problem: Consider
the functional

R c ( f , g,π):= c(x, y)π(dx, dy) , (8.6)
X×Y
where π is a bounded nonnegative Borel measure on X × Y with marginals f
and g, i.e. loosly speaking

π(dx, y) = g(y) (8.7)
X

π(x, dy) = f (x) (8.8)
Y
and minimize R c (f , g,π) over all those measures π:

P c ( f , g):= min R c ( f , g,π) . (8.9)

In fact the functional R c is linear in π and there is enough compactness to
proof that minimizing sequences converge to a minimizer. But how are these
two problems related? First of all, we note that for all admissible transportation
maps S the measure

π(x, y):= f (x)δ y − S(x) (8.10)
satisfies (8.7) and (8.8). However, generally, a minimizer π of (8.9) may not be of
the form (8.10) such that it does NOT in general correspond to a transportation
map and thus to a solution of the Monge problem.
Now let us consider the case, where f and g are absolutely continuous with
n
respecttotheLebesguemeasureonR ,representedbysmoothfunctions(which,
sloppily,wedenotebythe same symbols) of compactsupports X and Y,resp.,
and that the transportation cost is given by the quadratic function (8.4). In this
case the problem of constructing an optimal transportation plan was basically
3
resolved by Yann Brenier in [3] who proved a striking polar decomposition
theorem of smooth vector fields as composition of a gradient map (of a convex
scalar potential) and a Lebesgue measure preserving map. This very remarkable
3
http://math1.unice.fr/∼brenier