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

3.1. THE DIMENSION OF THE MATHER QUOTIENT 37
for any x, y, z ∈ M, and therefore it also satisfies the triangle inequality
h(x, z) ≤ h(x, y) + h(y, z).
Moreover, given a weak KAM solution u, we have
u(y) − u(x) ≤ h(x, y), ∀ x, y ∈ M.
In particular, we have h > −∞ everywhere. It follows, from the triangle inequality, that the
function h is either identically +∞ or it is finite everywhere. If M is compact, h is finite
everywhere. In addition, if h is finite, then for each x ∈ M the function hx(·) = h(x, ·) is a
critical viscosity solution. The projected Aubry set A is defined by
A = {x ∈ M | h(x, x) = 0}.
As done by Mather (see [60, page 1370]), one can symmetrize h to define the function δM :
M × M → R by
∀x, y ∈ M, δM(x, y) = h(x, y) + h(y, x).
Since h satisfies the triangle inequality and h(x, x) ≥ 0 everywhere, the function δM is
symmetric, everywhere nonnegative and satisfies the triangle inequality. The restriction δM :
A × A → R is a genuine semi-distance on the projected Aubry set. We call this function δM
the Mather semi-distance (even when we consider it on M rather than on A). We define the
Mather quotient (AM, δM) to be the metric space obtained by identifying two points x, y ∈ A if
their semi-distance δM(x, y) vanishes (we mention that this is set is also called quotient Aubry
set). When we consider δM on the quotient space AM we will call it the Mather distance.
3.1 The dimension of the Mather quotient
In [62], Mather formulated the following problem:
Mather’s Problem. If L is C ∞ , is the set AM totally disconnected for the topology of δM,
i.e. is each connected component of AM reduced to a single point?
In [61], Mather brought a positive answer to that problem in low dimension. More precisely,
he proved that if M has dimension two, or if the Lagrangian is the kinetic energy associated to
a Riemannian metric on M in dimension ≤ 3, then the Mather quotient is totally disconnected.
Mather mentioned in [62, page 1668] that it would be even more interesting to be able to prove
that the Mather quotient has vanishing one-dimensional Hausdorff measure, because this implies
the upper semi-continuity of the mapping H ↦→ A.
In [9], in a joint work with Albert Fathi and Ludovic, we were able to show that the vanishing
of the one-dimensional Hausdorff measure of the Mather quotient is satisfied under various
assumptions. Let us state our results.
Theorem 3.1.1 If dim M = 1, 2 and H of class C 2 or dim M = 3 and H of class C k,1 with
k ≥ 3, then the Mather quotient (AM, δM) has vanishing one-dimensional Hausdorff measure.