Definition 4.10.1 (Reversible Lagrangian). The Lagrangian L is said to be reversible if it satisfies L(x, −v) = L(x, v), for each (x, v) ∈ T M .

Example 4.10.2. Let g be a Riemannian metric on M , we denote
by k · k_{x} the norm deduced from g on T_{x}M . If V : M → R is C^{2},
the Lagrangian L defined by L(x, v) = ^{1}_{2}kvk^{2}_{x}− V (x) is reversible.

Proposition 4.10.3. For a reversible Lagrangian L, we have

−c[0] = inf

x∈ML(x, 0) = inf

(x,v)∈T ML(x, v).

Moreover ˜M0 = {(x, 0) | L(x, 0) = −c[0]}.

Proof. By the strict convexity and the superlinearity of L in the
fibers of the tangent bundle T M , we have L(x, 0) = inf_{v∈T}_{x}_{M}L(x, v),
for all x ∈ M . Let us set k = inf_{x∈M}L(x, 0) = inf_{(x,v)∈T M}L(x, v).

Since −c[0] = infR

L dµ, where the infimum is taken over all Borel
probability measures on T M invariant under the flow φ_{t}, we
ob-tain k ≤ −c[0]. Let then x_{0} ∈ M be such that L(x_{0}, 0) = k,
the constant curve ] − ∞, +∞[→ M, t 7→ x_{0} is a minimizing
ex-tremal curve. Consequently φ_{t}(x_{0}, 0) = (x_{0}, 0) and the Dirac mass
δ_{(x}_{0}_{,0)} is invariant by φ_{t}, butR

L dδ_{(x}_{0}_{,0)}= k. Therefore −c[0] = k
and (x0, 0) ∈ ˜M0. Let µ be a Borel probability measure on T M
such thatR

T ML dµ = −c[0]. Since −c[0] = inf_{T M}L, we
necessar-ily have L(x, v) = inf_{T M}L on the support of µ. It follows that
supp(µ) ⊂ {(x, 0) | L(x, 0) = −c[0]}.

We then consider the case where M is the circle T = R/Z.

We identify the tangent bundle T T with T × R. As a
Lagran-gian L we take one defined by L(x, v) = ^{1}_{2}v^{2} − V (x), where
V : T → R is C^{2}. We thus have −c[0] = inf^{T}_{×R}L = − sup V ,
hence c[0] = sup V . Let us identify T^{∗}T with T × R. The
Hamil-tonian H is given by H(x, p) = ^{1}_{2}p^{2}+ V (x). The differential
equa-tion on T^{∗}T which defines the flow φ_{t} is given by ˙x = p and

˙p = −V^{′}(x). If u_{−} ∈ S−, the compact subset Graph(du_{−}) is
con-tained in the set H^{−1}(c[0]) = {(x, p) | p = ±p

sup V − V (x)}.

We strongly encourage the reader to do some drawings FAIRE

DES DESSINS of the situation in R × R, the universal cover
of T × R. To describe u_{−} completely let us consider the case
where V reaches its maximum only at 0. In this case the set
H^{−1}(c[0]) consists of three orbits of φ^{∗}_{t}, namely the fixed point
(0, 0), the orbit O_{+} = {(x,p

sup V − V (x)) | x 6= 0} and the orbit O− = {(x, −p

sup V − V (x)) | x 6= 0}. On O_{+}the direction of the
increasing t is that of the increasing x (we identify in a natural way
T\ 0 with ]0, 1[). On O_{−}the direction of the increasing t is that of

It follows that there is a point x0 such that Graph(du−) is the union of (0, 0) and the two sets {(y,p

sup V − V (y)) | y ∈]0, x_{0}]}

and {(y, −p

sup V − V (y)) | y ∈ [x_{0}, 1[}. Moreover, since the
function u− is defined on T, we have limx→1u−(x) = u−(0) and
thus the integral on ]0, 1[ of the derivative of u_{−} must be 0. This
gives the relation

This equality determines completely a unique point x_{0}, since sup V −
V (x) > 0 for x ∈]0, 1[. In this case, we see that u− is unique up
to an additive constant and that

u−(x) =

Exercise 4.10.4. 1) If V : T → R reaches its maximum exactly
n times, show that the solutions u_{−} depend on n real parameters,
one of these parameters being an additive constant.

2) Describe the Mather function α : H^{1}(T, R) → R, Ω 7→ c[Ω].

3) If ω is a closed differential 1-form on T, describe the
func-tion u^{ω}_{−} for the Lagrangian L_{ω} defined by L(x, v) = ^{1}_{2}v^{2}− V (x) −
ω_{x}(v).

### Chapter 5

## Conjugate Weak KAM Solutions

In this chapter, as in the previous ones, we denote by M a
com-pact and connected manifold. The projection of T M on M is
denoted by π : T M → M . We suppose given a C^{r} Lagrangian
L : T M → R, with r ≥ 2, such that, for each (x, v) ∈ T M , the
second vertical derivative ^{∂}_{∂v}^{2}^{L}2(x, v) is definite > 0 as a quadratic
form, and that L is superlinear in each fiber of the tangent bundle
π : T M → M . We will also endow M with a fixed Riemannian
metric. We denote by d the distance on M associated with this
Riemannian metric. If x ∈ M , the norm k · k_{x} on T_{x}M is the one
induced by this same Riemannian metric.

### 5.1 Conjugate Weak KAM Solutions

We start with the following lemma

Lemma 5.1.1. If u ≺ L + c[0], then we have

∀x ∈ M0, ∀t ≥ 0, u(x) = T_{t}^{−}u(x) + c[0]t = T_{t}^{+}u(x) − c[0]t.

Proof. Since u ≺ L + c[0], we have u ≤ T_{t}^{−}u + c[0]t and u ≥
T_{t}^{+}u − c[0]t. We consider the point (x, v) ∈ ˜M0 above x. Let us
note by γ : ] − ∞, +∞[→ M the extremal curve s 7→ π(φ_{s}(x, v)).

167

By lemma 4.8.2, for each t ≥ 0, we have u(γ(0)) − u(γ(−t)) ≤

Z 0

−t

L(γ(s), ˙γ(s)) ds + c[0]t, u(γ(t)) − u(γ(0)) ≤

Z t 0

L(γ(s), ˙γ(s)) ds + c[0]t.

Since γ(0) = x, we obtain the inequalities u(x) ≥ T_{t}^{−}u(x) + c[0]t
and u(x) ≤ T_{t}^{+}u(x) − c[0]t.

Theorem 5.1.2 (Existence of Conjugate Pairs). If u : M → R
is a function such that u ≺ L + c[0], then, there exists a unique
function u_{−}∈ S−(resp. u_{+}∈ S+) with u = u_{−}(resp. u = u_{+}) on
the projected Mather set M_{0}. These functions verify the following
properties

(1) we have u_{+} ≤ u ≤ u−;

(2) if u^{1}_{−}∈ S− (resp. u^{1}_{+}∈ S+) verifies u ≤ u^{1}_{−} (resp. u^{1}_{+}≤ u),
then u_{−}≤ u^{1}_{−} (resp. u^{1}_{+} ≤ u_{+});

(3) We have u_{−}= lim_{t→+∞}T_{t}^{−}u+c[0]t and u_{+}= lim_{t→+∞}T_{t}^{+}u−

c[0]t, the convergence being uniform on M .

Proof. It will be simpler to consider the modified semigroup ˆT_{t}^{−}v =
T_{t}^{−}v + c[0]t. The elements of S_{−} are precisely the fixed points of
the semigroup ˆT_{t}^{−}. The condition u ≺ L + c[0] is equivalent to
u ≤ ˆT_{t}^{−}u. As ˆT_{t}^{−} preserves the order, we see that ˆT_{t}^{−}u ≤ u^{1}_{−} for
each u^{1}_{−} ∈ S_{−} satisfying u ≤ u^{1}_{−}. As ˆT_{t}^{−}u = u on the projected
Mather set M_{0}, it then remains to show that ˆT_{t}^{−}u is uniformly
convergent for t → ∞. However, we have ˆT_{t}^{−}u ≤ ˆT_{t+s}^{−} u, if s ≥ 0,
because this is true for t = 0 and the semigroup ˆT_{t}^{−} preserves the
order. Since for t ≥ 1 the family of maps ˆT_{t}^{−}u is equi-Lipschitzian,
it is enough to see that this family of maps is uniformly bounded.

To show this uniform boundedness, we fix u^{0}_{−} ∈ S−, by
compact-ness of M , there exists k ∈ R such that u ≤ u^{0}_{−}+ k. By what was
already shown, we have ˆT_{t}^{−}u ≤ u^{0}_{−}+ k.

Corollary 5.1.3. For any function u_{−} ∈ S− (resp. u_{+} ∈ S+),
there exists one and only one function of u_{+}∈ S+(resp. u_{−}∈ S−)
satisfying u_{+}= u_{−}on M_{0}. Moreover, we have u_{+}≤ u_{−}on all M .

169
Definition 5.1.4 (Conjugate Functions). A pair of functions (u_{−}, u_{+})
is said to be conjugate if u_{−}∈ S_{−}, u_{+}∈ S_{+} and u_{−}= u_{+} on M_{0}.
We will denote by D the set formed by the differences u_{−}− u+ of
pairs (u_{−}, u_{+}) of conjugate functions.

The following lemma will be useful in the sequel.

Lemma 5.1.5 (Compactness of the Differences). All the functions
in D are ≥ 0. Moreover, the subset D is compact in C^{0}(M, R) for
the topology of uniform convergence.

Proof. If u_{−} and u_{+} are conjugate, we then know that u_{+} ≤ u−

and thus u_{−} − u_{+} ≥ 0. If we fix x_{0} ∈ M , the set S_{−}^{x}^{0} = {u_{−} |
u_{−}(x_{0}) = 0} (resp. S_{+}^{x}^{0} = {u_{+} | u+(x_{0}) = 0}) is compact, since it
is a family of equi-Lipschitzian functions on the compact manifold
M which all vanishes at the point x_{0}. However, for c ∈ R, it is
obvious that the pair (u_{−}, u_{+}) is conjugate if and only if the pair
(u_{−}+ c, u_{+}+ c) is conjugate. We conclude that D is the subset
of the compact subset S_{−}^{x}^{0} − S_{+}^{x}^{0} formed by the functions which
vanish on M_{0}.

Corollary 5.1.6. Let us suppose that all the functions u_{−} ∈ S−

are C^{1} (what is equivalent to S_{−} = S_{+}). Then, two arbitrary
functions in S_{−} differ by a constant.

Proof. Conjugate functions are then equal, because the C^{1}
func-tions contained in S− or S+ are also in S−∩ S+ by 4.7.8.
Sup-pose then that u^{1}_{−} and u^{2}_{−} are two functions in S_{−}. We of course
do have u = (u^{1}_{−}+ u^{2}_{−})/2 ≺ L + c[0]. By the Theorem of
Ex-istence of Conjugate Pairs 5.1.2, we can find a pair of
conju-gate functions (u_{−}, u_{+}) with u_{+} ≤ u ≤ u_{−}. As conjugate
func-tions are equal, we have u = (u^{1}_{−} + u^{2}_{−})/2 ∈ S_{−}. The three
functions u, u^{1}_{−} and u^{2}_{−} are C^{1} and in S_{−}, we must then have
H(x, d_{x}(u^{1}_{−}+ u^{2}_{−})/2) = H(x, d_{x}u^{1}_{−}) = H(x, d_{x}u^{2}_{−}) = c[0], for each
x ∈ M . This is compatible with the strict convexity of H in fibers
of T^{∗}M only if d_{x}u^{1}_{−}= d_{x}u^{2}_{−}, for each x ∈ M .