We introduce a semigroup of non-linear operators (T_{t}^{−})_{t≥0} from
C^{0}(M, R) into itself. This semigroup is well-known in PDE and in
Calculus of Variations, it is called the Lax-Oleinik semigroup. To
define it let us fix u ∈ C^{0}(M, R) and t > 0. For x ∈ M , we set

T_{t}^{−}u(x) = inf

γ {u(γ(0)) + Z t

0

L(γ(s), ˙γ(s)) ds},

where the infimum is taken over all the absolutely continuous
curves γ : [0, t] → M such that γ(t) = x. Let us notice that
T_{t}^{−}u(x) ∈ R, because, by compactness of M and superlinearity of
L, we can find a lower bound ℓ_{0} of L on T M and we have

T_{t}^{−}u(x) ≥ ℓ0t − kuk∞,

where kuk_{∞} = sup_{x∈M}|u(x)| is the L^{∞} norm of the continuous
function u, which is of course finite by the compactness of M .
Lemma 4.4.1. If t > 0, u ∈ C^{0}(M, R) and x ∈ M are given, there
exists an extremal curve γ : [0, t] → M such that γ(t) = x and

T_{t}^{−}u(x) = u(γ(0)) +
Z t

0

L(γ(s), ˙γ(s)) ds.

Such an extremal curve γ minimizes the action among all
abso-lutely continuous curves γ_{1} : [0, t] → M with γ_{1}(0) = γ(0) and
γ_{1}(t) = x. In particular, we have

∀s ∈ [0, t], (γ(s), ˙γ(s)) ∈ Kt,

where K_{t} ⊂ T M is the compact set given by the A Priori
Com-pactness Corollary 4.3.2.

Proof. Let us fix y ∈ M and denote by γ_{y} : [0, t] → M an
ex-tremal curve minimizing the action among all absolutely
continu-ous curves γ : [0, t] → M such that γ(t) = x and γ(0) = y. We
have

T_{t}^{−}u(x) = inf

y∈M{u(y) + Z t

0

L(φ_{s}(y, ˙γ_{y}(0)) ds}.

133
By the a priori compactness given by corollary 4.3.2, the points
(y, ˙γ_{y}(0)) are all in the same compact subset K_{t} of T M . We can
then find a sequence of y_{n}∈ M such that

u(y_{n})+

Z t 0

L[φ_{s}(y_{n}, ˙γ_{y}_{n}(0))] ds → T_{t}^{−}u(x)and (y_{n}, ˙γ_{y}_{n}(0)) → (y_{∞}, v_{∞}).

By continuity of u, L and φ_{t}, we have
T_{t}^{−}u(x) = u(y_{∞}) +

Z t 0

L[φ_{s}(y_{∞}, v_{∞}))] ds.

The fact that γ(s) = πφ_{s}(x_{∞}, v_{∞}) minimizes the action is obvious
from the definition of T_{t}^{−}.

Since the extremal curves are C^{r}, with r ≥ 2, we can, in the
def-inition of semigroup T_{t}^{−}, replace the absolutely continuous curves
by (continuous) piecewise C^{1} (resp. of class C^{1} or even C^{r}) curves
without changing the value of T_{t}^{−}u(x).

It is probably difficult to find out when the next lemma ap-peared, in some of its forms, for the first time in the literature.

It has certainly been known for some time now, see for example [Fle69, Theorem1, page 518].

Lemma 4.4.2. For each t > 0, there exists a constant κ_{t} such
that T_{t}^{−}u : M → R is κ_{t}-Lipschitzian for each u ∈ C^{0}(M, R).

Proof. Let us consider ¯B(0, 3) the closed ball of center 0 and radius
3 in the Euclidean space R^{k}, where k is the dimension of M . By
compactness of M , we can find a finite number of coordinate charts
ϕ_{i} : ¯B(0, 3) → M, i = 1, . . . , p, such that M = Sp

i=1ϕ_{i}( ˚B(0, 1)).

We denote by η > 0, a constant such that

∀i = 1, . . . , p, ∀x, x^{′} ∈ M, d(x, x^{′}) ≤ η and x ∈ ϕ_{i} B(0, 1)¯

⇒
x^{′} ∈ ϕi B(0, 2)˚

and kϕ^{−1}_{i} (x^{′}) − ϕ^{−1}_{i} (x)k ≤ 1,
where the norm k · k is the Euclidean norm. Let us denote by
Kt the compact set obtained from corollary 4.3.2. We can find a
constant A < +∞ such that

∀(x, v) ∈ K_{t}, kvk_{x} ≤ A.

In the remaining part of the proof, we set ǫ = min t, η

A

.

In the same way by the compactness of K_{t}, we can find a constant
B < +∞ such that

∀x ∈ ¯B(0, 3), ∀v ∈ R^{k}, T ϕ_{i}(x, v) ∈ K_{t}⇒ kvk ≤ B.

Let us then give a point x ∈ M and suppose that i ∈ {1, . . . , p}

is such that x ∈ ϕ_{i} B(0, 1)˚

. Fix u ∈ C^{0}(M, R), we can find a
minimizing extremal curve γ : [0, t] → M such that γ(t) = x and

T_{t}^{−}u(x) = u(γ(0)) +
Z t

0

L(γ(s), ˙γ(s)) ds. (*)
By the a priori compactness given by corollary 4.3.2, we have
(γ(s), ˙γ(s)) ∈ K_{t}, for each s ∈ [0, t]. Consequently, we obtain
d(γ(t), γ(t^{′})) ≤ A|t − t^{′}|. In particular, since x ∈ ϕi B(0, 1)˚

and by the choice of ǫ, we find that

γ([t − ǫ, t]) ⊂ ϕ ˚B(0, 2) .

We can then define the curve ˜γ : [t − ǫ, t] → ˚B(0, 2) by ϕ_{i}(˜γ(s)) =
γ(s). If d(x, y) ≤ η, there is a unique ˜y ∈ ˚B(0, 2) such that
ϕ_{i}(˜y) = y. By the definition of η, we also have k˜y − ˜xk ≤ 1. Let
us define the curve ˜γ_{y} : [t − ǫ, t] → ˚B(0, 3) by

˜

γy(s) = s − (t − ǫ)

ǫ (˜y − ˜x) + ˜γ(s).

This curve connects the point ˜γ(t − ǫ) to the point ˜y, we can,
then, define the curve γ_{y} : [0, t] → M by γ_{y} = γ on [0, t − ǫ] and
γ_{y} = ϕ_{i}◦ ˜γ_{y} on [t − ǫ, t]. The curve γ_{y} is continuous on the interval
[0, t], moreover, it of class C^{r}on each of the two intervals [0, t − ǫ]

and [t − ǫ, t]. We of course have γ_{x}= γ. As γ_{y}(s) is equal to γ(0),
for s = 0, and is equal to y, for s = t, we have

T_{t}^{−}u(y) ≤ u(γ(0)) +
Z t

0

L(γ_{y}(s), ˙γ_{y}(s)) ds.

135
Subtracting the equality (∗) above from this inequality and using
the fact that γ_{y} = γ on [0, t − ǫ], we find

T_{t}^{−}u(y) − T_{t}^{−}u(x) ≤
Z t

t−ǫ

L(γ_{y}(s), ˙γ_{y}(s)) ds −
Z t

t−ǫ

L(γ(s), ˙γ(s)) ds.

We will use the coordinate chart ϕ_{i} to estimate the right-hand
side of this inequality. For that, it is convenient to consider the
Lagrangian ˜L : ¯B(0, 3) × R^{k}→ R defined by

L(x, v) = L ϕ˜ _{i}(x), Dϕ_{i}(x)[v]
.

This Lagrangian ˜L is of class C^{r} on ¯B(0, 3) × R^{k}. Using this
Lagrangian, we have

T_{t}^{−}u(y) − T_{t}^{−}u(x) ≤
Z _{t}

t−ǫ

L(˜˜ γ_{y}(s), ˙˜γ_{y}(s)) ds −
Z _{t}

t−ǫ

L(˜γ(s), ˙˜γ(s)) ds.

By the choice of B, we have k ˙˜γ(s)k ≤ B, for each s ∈ [t − ǫ, t].

By the definition of ˜γ_{y} and the fact that k˜y − ˜xk ≤ 1, we obtain

∀s ∈ [t − ǫ, t], k ˙˜γ_{y}(s)k ≤ k˜y − ˜xk

ǫ + B

≤ B + ǫ^{−1}.
Since ˜L is C^{1} and the set

E_{B,ǫ}= {(z, v) ∈ ¯B(0, 3) × R^{k}| kvk ≤ B + ǫ^{−1}}

is compact, we see that there exists a constant C_{i}, which depends
only on the restriction of the derivative of ˜L on this set E_{B,ǫ}, such
that

∀(z, v), (z^{′}, v^{′}) ∈ E_{B,ǫ}, | ˜L(z, v)− ˜L(z^{′}, v^{′})| ≤ C_{i}max(kz−z^{′}k, kv−v^{′}k).

Since γ_{y}(s)−γ(s) = ^{s−(t−ǫ)}_{ǫ} (˜y− ˜x) and the two points (˜γ_{y}(s), ˙˜γ_{y}(s))
and (˜γ(s), ˙˜γ(s)) are in E_{B,ǫ}, we find

|L(˜γ_{y}(s), ˙˜γ_{y}(s)) − L(˜γ(s), ˙˜γ(s))| ≤ C_{i}max

ks − (t − ǫ)

ǫ (˜y − ˜x)k, k1

ǫ(˜y − ˜x)k

≤ Cimax(1,1

ǫ)k˜y − ˜xk.

By integration on the interval [t − ǫ, t], it follows that
T_{t}^{−}u(y) − T_{t}^{−}u(x) ≤ C_{i}max(1,1

ǫ)k˜y − ˜xk.

Since ϕ_{i}is a diffeomorphism of class C^{∞}, its inverse is Lipschitzian
on the compact subset ϕ_{i} B(0, 3)¯

. We then see that there exists
a constant ˜C_{i}, independent of x, y and u and such that

x ∈ ϕ_{i} B(0, 1)˚

and d(x, y) ≤ η ⇒ T_{t}^{−}u(y) − T_{t}^{−}u(x) ≤ ˜C_{i}d(x, y).

Setting κ_{t}= max^{p}_{i=1}C˜_{i}, we find that we have

d(x, y) ≤ η ⇒ T_{t}^{−}u(y) − T_{t}^{−}u(x) ≤ κ_{t}d(x, y).

If x, y are two arbitrary points, and γ : [0, d(x, y)] is a geodesic of
length d(x, y), parameterized by arclength and connecting x to y,
we can find a finite sequence t_{0} = 0 ≤ t_{1}≤ · · · ≤ tℓ= d(x, y) with
ti+1− ti ≤ η. Since d(γ(ti+1), γ(ti)) = ti+1− ti≤ η, we obtain

∀i ∈ {0, 1, . . . , ℓ − 1}, T_{t}^{−}u(γ(t_{i+1})) − T_{t}^{−}u(γ(t_{i})) ≤ κ_{t}(t_{i+1}− t_{i}).

Adding these inequalities, we find

T_{t}^{−}u(y) − T_{t}^{−}u(x) ≤ κ_{t}d(x, y).

We finish the proof by exchanging the roles of x and y.

We recall the following well-known definition

Definition 4.4.3 (Non-expansive Map). A map ϕ : X → Y , between the metric spaces X and Y , is said to be non-expansive if it is Lipschitzian with a Lipschitz constant ≤ 1.

Corollary 4.4.4. (1) Each T_{t}^{−} maps C^{0}(M, R) into itself.

(2) (Semigroup Property) We have T_{t+t}^{−} ′ = T_{t}^{−}◦ T_{t}^{−}^{′}, for
each t, t^{′} > 0.

(3) (Monotony) For each u, v ∈ C^{0}(M, R) and all t > 0, we
have

u ≤ v ⇒ T_{t}^{−}u ≤ T_{t}^{−}v.

(4) If c is a constant and u ∈ C^{0}(M, R), we have T_{t}^{−}(c + u) =
c + T_{t}^{−}u.

137
(5) (Non-expansiveness) the maps T_{t}^{−} are non-expansive

∀u, v ∈ C^{0}(M, R), ∀t ≥ 0, kT_{t}^{−}u − T_{t}^{−}vk_{∞}≤ ku − vk∞.
(6) For each u ∈ C^{0}(M, R), we have lim_{t→0}T_{t}^{−}u = u.

(7) For each u ∈ C^{0}(M, R), the map t 7→ T_{t}^{−}u is uniformly
continuous.

(8) For each u ∈ C^{0}(M, R), the function (t, x) 7→ T_{t}^{−}u(x) is
con-tinuous on [0, +∞[×M and locally Lipschitz on ]0, +∞[×M . In
fact, for each t_{0}> 0, the family of functions (t, x) 7→ T_{t}^{−}u(x), u ∈
C^{0}(M, R), is equi-Lipschitzian on [t0, +∞[×M .

Proof. Assertion (1) is a consequence of Lemma 4.4.2 above. The
assertions (2), (3) and (4) result from the definition of T_{t}^{−}.

To show (5), we notice that −ku − vk_{∞}+ v ≤ u ≤ ku − vk_{∞}+ v
and we apply (3) and (4).

As the C^{1} maps form a dense subset of C^{0}(M, R) in the
topol-ogy of uniform convergence, it is enough by (5) to show property
(6) when u is Lipschitz. We denote by K the Lipschitz constant
of u. By the compactness of M and the superlinearity of L, there
is a constant CK such that

∀(x, v) ∈ T M, L(x, v) ≥ Kkvkx+ C_{K}.
It follows that for every curve γ : [0, t] → M , we have

Z t 0

L(γ(s), ˙γ(s)) ds ≥ Kd(γ(0), γ(t)) + C_{K}t.

Since the Lipschitz constant of u is K, we conclude that Z t

0

L(γ(s), ˙γ(s)) ds + u(γ(0)) ≥ u(γ(t)) + C_{K}t,
which gives

T_{t}^{−}u(x) ≥ u(x) + C_{K}t.

Moreover, using the constant curve γ_{x} : [0, t] → M, s 7→ x, we
obtain

T_{t}^{−}u(x) ≤ u(x) + L(x, 0)t.

Finally, if we set A_{0} = max_{x∈M}L(x, 0), we found that
kT_{t}^{−}u − uk_{∞}≤ t max(C_{K}A_{0}),

which does tend to 0, when t tends to 0.

To show (7), we notice that by the property (2) of semigroup, we have

kT_{t}^{−}^{′}u − T_{t}^{−}uk_{∞}≤ kT_{|t}^{−}′−t|u − uk_{∞},
and we apply (6).

To prove (8), we remark that the continuity of (t, x) 7→ T_{t}^{−}u(x)
follows from (1) and (7). To prove the equi-Lipschitzianity, we
fix t_{0} > 0. By Lemma 4.4.2, we know that there exists a finite
constant K(t_{0}) such that T_{t}^{−}u is Lipschitz with Lipschitz constant

≤ K(t0), for each u ∈ C^{0}(M, R) and each t ≥ t_{0}. It follows from
the semi-group property and the proof of (6) that

kT_{t}^{−}^{′} u − T_{t}^{−}uk_{∞}≤ (t^{′}− t) max(C_{K(t}_{0}_{)}, A_{0}),

for all t^{′}≥ t ≥ t0. It is then easy to check that (t, x) 7→ T_{t}^{−}u(x) is
Lipschitz on [t_{0}, +∞[×M , with Lipschitz constant ≤ max(C_{K(t}_{0}_{)}, A_{0})+

K(t_{0}), for anyone of the standard metrics on the product R ×
M .

Before proving the weak KAM Theorem, we will need to recall some fixed point theorems. We leave this as an exercise, see also [GK90, Theorem 3.1, page 28].

Exercise 4.4.5. 1) Let E be a normed space and K ⊂ E a com-pact convex subset. We suppose that the map ϕ : K → K is non-expansive. Show that ϕ has a fixed point. [Hint: Reduce first to the case when 0 ∈ K, and then consider x 7→ λϕ(x), with λ ∈]0, 1[.]

2) Let E be a Banach space and let C ⊂ E be a compact subset.

Show that the closed convex envelope of C in E is itself compact.

[Hint: It is enough to show that, for each ǫ > 0, we can cover the closed convex envelope of C by a finite number of balls of radius ǫ.]

3) Let E be a Banach space. If ϕ : E → E is a non-expansive map such that ϕ(E) has a relatively compact image in E, then the ϕ map admits a fixed point. [Hint: Take a compact convex subset containing the image of ϕ.]

4) Let E be a Banach space and ϕ_{t}: E → E be a family of maps
defined for t ∈ [0, ∞[. We suppose that the following conditions
are satisfied

139

• For each t, t^{′} ∈ [0, ∞[, we have ϕ_{t+t}^{′} = ϕ_{t}◦ ϕ_{t}^{′}.

• For each t ∈ [0, ∞[, the map ϕ_{t} is non-expansive.

• For each t > 0, the image ϕt(E) is relatively compact in E.

• For each x ∈ E, the map t 7→ ϕt(x) is continuous on [0, +∞[.

Show then that the maps ϕ_{t} have a common fixed point. [Hint:

A fixed point of ϕ_{t} is a fixed point of ϕ_{kt} for each integer k ≥ 1.

Show then that the maps ϕ_{1/2}^{n}, with n ∈ N, admit a common fixed
point.]

We notice that we can use Brouwer’s Fixed Point Theorem (instead of Banach’s Fixed Point Theorem) and an approximation technique to show that the result established in part 1) of the exercise above remains true if ϕ is merely continuous. This is the Schauder-Tykhonov Theorem, see [Dug66, Theorems 2.2 and 3.2, pages 414 and 415] or [DG82, Theorem 2.3, page 74]. It follows that the statements in parts 3) and 4) are also valid when the involved maps are merely continuous.

Theorem 4.4.6 (Weak KAM). There exists a Lipschitz function
u_{−}: M → R and a constant c such that T_{t}^{−}u_{−}+ ct = u_{−}, for each
t ∈ [0, +∞[.

Proof. Let us denote by 1 the constant function equal to 1
every-where on M and consider the quotient E = C^{0}(M, R)/R.1. This
quotient space E is a Banach space for the quotient norm

k[u]k = inf

a∈Rku + a1k∞,

where [u] is the class in E of u ∈ C^{0}(M, R). Since T_{t}^{−}(u + a1) =
T_{t}^{−}(u) + a1, if a ∈ R, the maps T_{t}^{−} pass to the quotient to a
semigroup ¯T_{t}^{−} : E → E consisting of non-expansive maps. Since,
for each t > 0, the image of T_{t}^{−} is an equi-Lipschitzian family
of maps, Ascoli’s Theorem, see for example [Dug66, Theorem 6.4
page 267], , then shows shows that the image of ¯T_{t}^{−} is relatively
compact in E (exercise). Using part 4) in the exercise above, we
find a common fixed point for all the ¯T_{t}^{−}. We then deduce that
there exists u_{−} ∈ C^{0}(M, R) such that T_{t}^{−}u_{−} = u_{−}+ c_{t}, where c_{t}

is a constant. The semigroup property gives c_{t+t}^{′} = c_{t}+ c_{t}^{′}; since
t 7→ T_{t}^{−}u is continuous, we obtain c_{t} = −tc with c = −c_{1}. We
thus have T_{t}^{−}u_{−}+ ct = u_{−}.

The following lemma results immediately from the definitions.

Lemma 4.4.7. If u : M → R, we have u ≺ L + c on M if and
only if u ≤ ct + T_{t}^{−}u, for each t ≥ 0.

Proposition 4.4.8. Suppose that u ∈ C^{0}(M, R) and c ∈ R. We
have T_{t}^{−}u + ct = u, for each t ∈ [0, +∞[, if and only if u ≺ L + c
and for each x ∈ M , there exists a (u, L, c)-calibrated curve γ_{−}^{x} :
] − ∞, 0] → M such that γ^{x}_{−}(0) = x.

Proof. We suppose that T_{t}^{−}u + ct = u, for each t ∈ [0, +∞[. By
Lemma ?? above u ≺ L + c. It remains to show the existence
of γ_{−}^{x} : ] − ∞, 0] → M , for a given x ∈ M . We already know
that, for each t > 0, there exists a minimizing extremal curve
γ_{t}: [0, t] → M , with γ_{t}(t) = x and

u_{−}(x) − ct = T_{t}u_{−}(x) = u_{−}(γ_{t}(0)) +
Z t

0

L(γ_{t}(s), ˙γ_{t}(s)) ds.

We set ¯γ_{t}(s) = γ_{t}(s + t), this curve is parametrized by the interval
[−t, 0], is equal to x at 0, and satisfies

u_{−}(¯γ_{t}(0)) − u_{−}(¯γ_{t}(−t)) = ct +
Z 0

−t

L(¯γ_{t}(s), ˙¯γ_{t}(s)) ds.

It follows from Proposition 4.4.8 above that ¯γ_{t}is (u, L, c)-calibrated.

In particular, we have

∀t^{′}∈ [−t, 0], u−(x)−u−(¯γt(t^{′})) = −ct^{′}+
Z 0

t^{′}

L(¯γt(s), ˙¯γt(s)) ds. (*)
As the ¯γ_{t} are minimizing extremal curves, by the a priori
com-pactness given by corollary 4.3.2, there exists a compact subset
K1 ⊂ T M , such that

∀t ≥ 1, ∀s ∈ [−t, 0], (¯γ_{t}(s), ˙¯γ_{t}(s)) ∈ K_{1}.

Since the ¯γ_{t}are extremal curves, we have (¯γ_{t}(s), ˙¯γ_{t}(s)) = φ_{s}(¯γ_{t}(0), ˙¯γ_{t}(0)).

The points (¯γ_{t}(0), ˙¯γ_{t}(0)) are all in the compact subset K_{1}, we can

141
find a sequence t_{n}ր +∞ such that the sequence (¯γ_{t}_{n}(0), ˙¯γ_{t}_{n}(0)) =
(x, ˙¯γ_{t}_{n}(0)) tends to (x, v_{∞}), when n → +∞. The negative orbit
φ_{s}(x, v_{∞}), s ≤ 0 is of the form (γ_{−}^{x}(s), ˙γ_{−}^{x}(s)), where γ_{−}^{x} :]−∞, 0] →
M is an extremal curve with γ_{x}(0) = x. If t^{′} ∈]−∞, 0] is fixed, for n
large enough, the function s 7→ (¯γ_{t}_{n}(s), ˙¯γ_{t}_{n}(s)) = φ_{s}(¯γ_{t}_{n}(0), ˙¯γ_{t}_{n}(0))
is defined on [t^{′}, 0], and, by continuity of the Euler-Lagrange flow,
this sequence converges uniformly on the compact interval [t^{′}, 0]

to the map s 7→ φ_{s}(x, v_{∞}) = (γ_{x}(s), ˙γ_{x}(s)). We can then pass to
the limit in the equality (∗) to obtain

u_{−}(x) − u_{−}(γ_{−}(t^{′})) = −ct^{′}+
Z 0

t^{′}

L(γ_{−}^{x}(s), ˙γ_{−}^{x}(s)) ds.

Conversely, let us suppose that u ≺ L+c and that, for each x ∈ M ,
there exists a curve γ_{−}^{x} : ] − ∞, 0] → M , with γ_{−}^{x}(0) = x, and such
that for each t ∈ [0, +∞[

u_{−}(x) − u_{−}(γ_{−}^{x}(−t)) = ct +
Z 0

−t

L(γ_{−}^{x}(s), ˙γ_{−}^{x}(s)) ds.

Let us show that T_{t}^{−}u + ct = u, for each t ∈ [0, +∞[. If x ∈ M
and t > 0, we define the curve γ : [0, t] → M by γ(s) = γ_{−}^{x}(s − t).

It is not difficult to see that γ(t) = x and that

u_{−}(x) = ct +
Z t

0

L(γ(s), ˙γ(s)) ds + u_{−}(γ(0)).

It follows that T_{t}^{−}u(x) + ct ≤ u(x) and thus T_{t}^{−}u + ct ≤ u. The
inequality u ≤ T_{t}^{−}u + ct results from u ≺ L + c.

Corollary 4.4.9. If T_{t}^{−}u_{−}= u_{−}+ ct, then, we have

−c = inf

µ

Z

T M

L dµ,

where µ varies among Borel probability measures on T M invariant
by the Euler-Lagrange flow φ_{t}. This lower bound is in fact achieved
by a measure with compact support. In particular, the constant c
is unique.

Proof. If (x, v) ∈ T M , then γ : [0, +∞[→ M defined by γ(s) =
is integrable since it is bounded. Invariance of the measure by φ_{t}
gives

Z

T M

[u_{−}(πφ_{1}(x, v)) − u_{−}(π(x, v))] dµ(x, v) = 0,
from where, by integration of the inequality above, we obtain

0 ≤

Since L is bounded below and µ is a probability measure, we can apply Fubini Theorem to obtain

Since µ is a probability measure, this yields

−c ≤ Z

T M

L dµ.

It remains to see that the value −c is attained. For that, we fix
x ∈ M , and we take a curve γ_{x}^{−} :] − ∞, 0] → M with γ_{x}^{−}(0) = x

143
and the curve (γ_{x}^{−}(s), ˙γ_{x}^{−}(s)), s ≤ 0 is entirely contained in a
com-pact subset K_{1}of T M as given by corollary 4.3.2. Using Riesz
Rep-resentation Theorem [Rud87, Theorem 2.14, page 40], for t > 0,
we define a Borel probability measure µ_{t} on T M by

µ_{t}(θ) = 1
t

Z _{0}

−t

θ(φ_{s}(x, ˙γ_{x}^{−}(0))) ds,

for θ : T M → R a continuous function. All these probability
measures have their supports contained in the compact subset K1,
consequently, we can extract a sequence t_{n} ր +∞ such that µtn

converges weakly to a probability measure µ with support in K_{1}.
Weak convergence means that for each continuous function θ :
T M → R, we have
which is bounded by 2ku_{−}k∞. This does indeed show that for the
limit measure µ, we haveR

T ML dµ = −c.

In the second part of the previous proof, we have in fact shown the following proposition.

• for each s ≥ 0, we have φ−s(x, ˙γ_{x}^{−}(0)) = (γ_{x}^{−}(−s), ˙γ_{x}^{−}(−s));

• the α-limit set of the orbit of (x, ˙γ_{x}^{−}(0)) for φ_{s} is compact;

• there exists a Borel probability measure µ on T M , invariant
by φ_{t}, carried by the α-limit set of the orbit of (x, ˙γ^{−}_{x}(0)),
and such thatR

L dµ = −c.

We define c[0] ∈ R by

c[0] = − inf

µ

Z

T M

L dµ,

where the lower bound is taken with respect to all Borel probability
measures on T M invariant by the Euler-Lagrange flow. We will
use the notation c_{L}[0], if we want to specify the Lagrangian.

Definition 4.4.11 (Minimizing Measure). A measure µ on T M is said to be minimizing if it is Borel probability measure µ, invariant by the Euler-Lagrange flow, and

c[0] = − Z

T M

L dµ.

Exercise 4.4.12. Show that each Borel probability measure µ,
in-variant by the Euler-Lagrange flow, and whose support is contained
in the α-limit set of a trajectory of the form t 7→ (γ_{−}^{x}(t), ˙γ_{−}^{x}(t)),
must be minimizing.

Corollary 4.4.13. If a weak KAM solution u_{−} is differentiable
at x ∈ M , we have

H(x, d_{x}u_{−}) = c[0].

Proof. If u_{−} is a weak KAM solution, we have u_{−}≺ L + c[0] and
thus

H(x, d_{x}u_{−}) = sup

v∈TxM

d_{x}u_{−}(v) − L(x, v) ≤ c[0].

It remains to find v0 ∈ TxM such that dxu−(v0) = L(x, v0) + c[0].

For that, we pick extremal curves γ_{−}^{x} :]−∞, 0] such that γ_{−}^{x}(0) = x
and

∀t ≥ 0, u−(γ_{−}^{x}(0)) − u_{−}(γ^{x}_{−}(−t)) =
Z _{0}

−t

L(γ_{−}^{x}(s), ˙γ_{−}^{x}(s)) ds + c[0]t.

145

By dividing this equality by t > 0 and letting t tend to 0, we find
d_{x}u_{−}( ˙γ_{−}^{x}(0)) = L(x, ˙γ_{−}^{x}(0)) + c[0].