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

16 CHAPTER 1. THE OPTIMAL TRANSPORT PROBLEM
As another byproduct of our regularity result, we can prove the absolute continuity of Wasserstein
geodesics. This fact was stated as an open problem in the case of the Heisenberg group
[29], and solved in the Heisenberg group and in Alexandrov spaces in a joint work with Nicolas
Juillet [17]:
Theorem 1.3.3 With the same assumption of Theorem 1.3.1, there exists a unique Wasserstein
geodesic (µt) t∈[0,1] joining µ = µ0 to ν = µ1, which is given by µt := (Tt)#µ for t ∈ [0, 1], with
expx(−t dφ(x)) if x ∈ A,
Tt(x) :=
x if x ∈ M \ A.
Moreover, if Ω is totally geodesically convex, then µt is absolutely continuous for all t ∈ [0, 1).
1.4 Regularity of the optimal transport on Riemannian manifolds
Let (M, g) be a compact connected Riemannian manifold, let µ(dx) = f(x)vol(dx) and ν(dy) =
g(y)vol(dy) be probability measures on M, and consider the cost c(x, y) = 1
2 d(x, y)2 . Assume f
and g to be C ∞ and strictly positive on M. A natural question is whether the optimal map T
is smooth or not.
To understand a bit the problem, we start from the Jacobian equation
det(dxT ) =
f(x)volx
,
g(T (x))volT (x)
and the relation T (x) = expx ∇ϕ(x) . We now write a PDE for ϕ. Indeed, since
∇ϕ(x) + ∇xc(x, T (x)) = 0,
differentiating with respect to x and using the Jacobian equation we get
det ∇ 2 ϕ(x) + ∇ 2 xc
x, expx ∇ϕ(x) =
f(x)volx
g(T (x))vol det(d T (x) ∇ϕ(x) expx)
=: h(x, ∇ϕ(x)).
We see that ϕ solves a Monge-Ampère type equation with a perturbation ∇2 xc
x, expx ∇ϕ(x)
which is of first order in ϕ. Unfortunately, for Monge-Ampère type equations lower order terms
do matter, and it turns out that it is exactly the term ∇2 xc
x, expx ∇ϕ(x) which can create
obstractions to the smoothness.
In [63], the authors found a mysterious forth-order conditions on the cost functions, which
turned out to be sufficient to prove regularity results. The idea was to differentiate twice the
above PDE for ϕ in order to get a linear PDE for the second derivatives of ϕ, and then try to
prove an a priori estimate. In this computation, one ends up at a certain moment with a term
which needs to have a sign in order to make the equation elliptic. This term is what now is
called the Ma-Trudinger-Wang tensor (in short MTW tensor):
S (x,y)(ξ, η) := 3
(cij,rc
2
r,s cs,kl − cij,kl) ξ i ξ j η k η l , ξ ∈ TxM, η ∈ TyM.
ijklrs