Optimal transport, Euler equations, Mather and DiPerna-Lions theories

1.3. SUB-RIEMANNIAN MANIFOLDS 15
Theorem 1.3.1 Let µ and ν be (compactly supported) probability measures, with µ absolutely
continuous with respect to vol. Assume that there exists an open set Ω ⊂ M × M such that
supp(µ × ν) ⊂ Ω, and d2 SR is locally semiconcave (resp. locally Lipschitz) on Ω \ D. Then there
exists a function φ : M → R, and an open set A ⊂ M, such that φ is locally semiconcave inside
A, and the unique optimal transport map is given by
T (x) :=
expx(− dφ(x)) if x ∈ A,
x if x ∈ M \ A.
We see that we can recover more or less the standard Riemannian result, simply splitting
the transport map in two sets suitably chosen. The key points with respect to previous results
in the sub-Riemannian setting are:
1) We do not make any assumption of regularity on the sub-Riemannian distance on the
diagonal (all previous results assumed at least the function d2 SR to be Lipschitz on the
diagonal, like in the Heisenberg group).
2) We can prove a “second order regularity” of the function φ appearing in the formula for
the transport map (recall indeed that semiconcave function are twice differentiable a.e.)
In particular, thanks to 2), we can prove a weak regularity property of the optimal transport
map, as in the Riemannian case. This allows for instance to write (for the first time, to our
knowledge) a weak formulation of Monge-Ampère equation in a sub-Riemannian setting:
Theorem 1.3.2 With the same assumption of Theorem 1.3.1, the optimal transport map is
differentiable µ-a.e. inside A, it is approximately differentiable at µ-a.e. x. Moreover
Y (x) := d(exp x) −dφ(x) and H(x) := 1
2 Hess dSR(·, T (x)) 2 |z=x
exists for µ-a.e. x ∈ A, and the approximate differential of T is given by the formula
˜dxT
Y (x) H(x) − Hess φ(x) if x ∈ A,
=
Id if x ∈ M \ A,
where Id : TxM → TxM denotes the identity map.
Finally, assuming both µ and ν absolutely continuous with respect to the volume measure,
and denoting by f and g their respective density, the following Jacobian identity holds:
det ˜dxT
f(x)
= = 0 µ-a.e. (1.3.3)
g(T (x))
In particular φ satisfies in a weak sense the Monge-Ampère type equation
det H(x) − Hess φ(x) =
f(x)
| det(Y (x))|g(T (x))
for µ-a.e. x ∈ A.