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

1.5. THE (ANISOTROPIC) ISOPERIMETRIC INEQUALITY 19
the action of O(n), or even of SO(n), and in fact it may even happen that PK(E) = PK(Rn \E),
provided K is not symmetric with respect to the origin. When K is the Euclidean unit ball
B = {x ∈ Rn : |x| < 1} of Rn , then ν∗ = 1 for every ν ∈ Sn−1 , and therefore PK(E) coincides
with the Euclidean perimeter of E.
Apart from its intrinsic geometric interest, the anisotropic perimeter PK arises as a model
for surface tension in the study of equilibrium configurations of solid crystals with sufficiently
small grains, and constitutes the basic model for surface energies in phase transitions. In the
former setting, one is naturally led to minimize PK(E) under a volume constraint. This is of
course equivalent to study the isoperimetric problem
PK(E)
inf
: 0 < |E| < ∞ , (1.5.2)
|E| (n−1)/n
where |E| is the Lebesgue measure of E. As conjectured by Wulff [75] back to 1901, the unique
minimizer (modulo the invariance group of the functional, that consists of translations and
scalings) is the set K itself. In particular the anisotropic isoperimetric inequality holds:
PK(E) ≥ n|K| 1/n |E| (n−1)/n , if |E| < ∞. (1.5.3)
It was Dinghas [50] to show how to derive (1.5.3) from the Brunn-Minkowski inequality
|E + F | 1/n ≥ |E| 1/n + |F | 1/n , ∀ E, F ⊆ R n . (1.5.4)
The formal argument is well known. Indeed, (1.5.4) implies that
|E + εK| − |E|
ε
≥ (|E|1/n + ε|K| 1/n ) n − |E|
, ∀ ε > 0.
ε
As ε → 0 + , the right hand side converges to n|K| 1/n |E| (n−1)/n , while, if E is regular enough,
the left hand side has PK(E) as its limit.
Gromov’s proof of the anisotropic isoperimetric inequality
Although Gromov’s proof [67] was originally based on the use of the Knothe map M between E
and K, his argument works with any other transport map having suitable structure properties,
like the Brenier map. This is a well-known, common feature of all the proofs of geometricfunctional
inequalities based on mass transportation [45]. However it seems that, in the study
of stability, Brenier map is more efficient.
We now want to give the proof of the anisotropic isoperimetric inequality, without caring
about regularity issues.
Let us apply Theorem 1.1.1 to the measures µ = 1
|E| χE dx, ν = 1
|K| χK dy. Then we know
that there exists a transport map T which takes E into K and such that
det ∇T = |K|
|E|
on E.