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8 Optimal Transportation and Monge–Ampère Equations
131

Fig. 8.2. ‘Deblais’, old fashioned Mass Transportation

where DS(x) denotes the Jacobian of the map S = S(x). Now let c = c(x, y)bethe
transportation cost (or work), given by a nonnegative measurable function c,
+
which maps X × Y → R .Wecan thinkofthe value c(x, y)asthe cost or work
it takes to move the mass point x in X into the point y in Y. Then the total
transportation cost (or work) is defined by:

C c (f , g; S):= c x, S(x) df . (8.2)

X
As already mentioned above, intuitively speaking, c typically is a function of
the Euclidean distance |x − y|. Actually, very important classical cases are

c(x, y):= |x − y| (8.3)

used by Monge, assuming that the transportation cost is equal to the distance of
a mass point before and after transportation, and the quadratic case

2
|x − y|
c(x, y):= . (8.4)
2
In non-standard applications as in urban transportation network planning
or in irrigation networks other, more complicated cost functions arise.
The Monge formulation of the optimal transportation problem reads:

O c (f , g) = inf C c ( f , g; S) , (8.5)