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8 Optimal Transportation and Monge–Ampère Equations
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theorem can be regarded as a nonlinear version of the Helmholtz decomposition
theorem, which additively decomposes a smooth vector field into a divergence
free vector field, tangential to the boundary of the domain, and a gradient
map. It turns out, by using a dual formulation by Kantorovich of the Monge
problem (which is a continuous version of linear programming) that the optimal
transportation map is the gradient of a convex potential, i.e.

S opt (x) = grad V(x), V convex on X . (8.11)

Since (8.1) implies (after a weak formulation using test functions), assuming
sufficient smoothness of S:

g S(x) det DS(x) = f (x), (8.12)
we conclude that V is a (weak) solution to the following Monge–Ampère equa-
tion:
2
g(grad V)det(D V) = f (x), x ∈ X (8.13)
grad V : X → Y
2
(D V stands for the Hessian of V). This links the Monge–Kantorovich mass
transportationtheorytotheareaofpartialdifferentialequations.Weremarkthat
the Monge–Ampère equation is a fully nonlinear elliptic differential equation,
which has only recently been investigated in a detailed mathematical way. In
4
particular we refer to the work of Luis Caffarelli [6], [7], [8], which presents
adeepregularitytheoryfortheMonge–Ampèreequation,basicallygivingresults
analogous to the Schauder theory for linear elliptic equations. Note that even
the interpretation of the equation (8.13) is not obvious since convex functions
2
in general do not have pointwise second derivatives, in full generality D V is –
due to convexity of V – a matrix of signed measures only!
The solution of the Monge–Kantorovich optimisation problem is even more
complicated if the cost function s = s(|x − y|) is not uniformy convex, e.g. in the
original case of Monge (8.3). Here we only mention that, again under the above
assumptions on the measures f and g, the optimal transportation map exists
and satisfies:

S opt (x) = x − a(x)grad u(x) , (8.14)

where u is again a scalar potential with |grad u(x)| = 1and a is nonnegative. We
referto[9] on howtorecover u and the distance a from the measures f and g.
Obviously, for more complex realistic applications the cost functional has to
be adapted, in particular when there are (geometrical or other) constraints on
the transportation trajectories. Heuristically speaking, the Monge–Kantorovich
problem corresponds to the case where all possible transportation roads exist
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http://rene.ma.utexas.edu/users/caffarel/