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

1.6. THE OPTIMAL PARTIAL TRANSPORT PROBLEM 25
- if K < 0,
S n−1
e −(n−1)GK(f) sinh(
|K|f)
n−1 |∇f|2
1 + ℓK(f) 4
|K|
f
≥ (nωn) 1/n
S n−1
2 dH n−1
e −nGK(f) sinh( |K|f)
n dH
|K|
n−1
(n−1)/n
. (1.5.13)
All these results show how optimal transport reveals to be an extremely powerful instrument
for (im)proving functional inequalities. One of my projects is to try to see how to apply these
strategy to other cases, for instance to improve log-Sobolev inequalities.
1.6 The optimal partial transport problem
The optimal partial transport problem is a variant of the classical optimal transport problem:
given two densities f and g, we want to transport a fraction m ∈ [0, min{fL1, gL1}] of
the mass of f onto g minimizing the transportation cost c(x, y) = |x − y| 2 . More precisely, let
f, g ∈ L1 (Rn ) be two nonnegative functions, and denote by Γ≤(f, g) the set of nonnegative Borel
measures on Rn ×Rn whose first and second marginals are dominated by f and g respectively. Fix
a certain amount m ∈ [0, min{fL1, gL1}] which represents the mass one wants to transport,
and consider the following partial transport problem:
minimize C(γ) := |x − y| 2 dγ(x, y)
R n ×R n
among all γ ∈ Γ≤(f, g) with dγ = m.
Using weak topologies, it is simple to prove existence of minimizers for any fixed amount
of mass m ∈ [0, min{f L 1, g L 1}]. We remark however that in general one cannot expect
uniqueness of minimizers: if m ≤
R n f ∧ g, any γ supported on the diagonal {x = y} with
marginals dominated by f ∧ g is a minimizer with zero cost. To ensure uniqueness, in [43]
Caffarelli and McCann assume f and g to have disjoint supports. Under this assumption they
are able to prove (as in the classical Monge-Kantorovich problem) that there exists a (unique)
convex function ϕ such that the unique minimizer is concentrated on the graph of ∇ϕ. This
ϕ is also shown to solve in a weak sense a Monge-Ampère double obstacle problem. Then,
strengthening the disjointness assumption into the hypothesis on the existence of a hyperplane
separating the supports of the two measures, they can prove a semiconvexity result on the free
boundaries. Furthermore, under some classical regularity assumptions on the measures and on
their supports, local C1,α regularity of ϕ and on the free boundaries of the active regions is
shown.
In [15], I studied what happens if one removes the disjointness assumption. Although minimizers
are non-unique for m <
Rn f ∧ g (but in this case the set of minimizers can be trivially
described), uniqueness holds for any m ≥
Rn f ∧ g. Moreover, exactly as in [43], the unique
minimizer is concentrated on the graph of the gradient of a convex function.