The Lax-Oleinik Semigroup - Weak KAM Theorem in Lagrangian Dynamics Seventh Preliminary Version

We introduce a semigroup of non-linear operators (T_t⁻)_t≥0 from C⁰(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 ∈ C⁰(M, R) and t > 0. For x ∈ M , we set

T_t⁻u(x) = inf

γ {u(γ(0)) + Z t

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 ℓ₀ 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 ∈ C⁰(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

L(γ(s), ˙γ(s)) ds.

Such an extremal curve γ minimizes the action among all abso-lutely continuous curves γ₁ : [0, t] → M with γ₁(0) = γ(0) and γ₁(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

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 C¹ (resp. of class C¹ 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 ∈ C⁰(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

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 ∈ C⁰(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

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^ron 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

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 C¹ 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_imax(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_imax

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_imax(1,1

ǫ)k˜y − ˜xk.

Since ϕ_iis 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_id(x, y).

Setting κ_t= max^p_i=1C˜_i, we find that we have

d(x, y) ≤ η ⇒ T_t⁻u(y) − T_t⁻u(x) ≤ κ_td(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 t₀ = 0 ≤ t₁≤ · · · ≤ 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_td(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 C⁰(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 ∈ C⁰(M, R) and all t > 0, we have

u ≤ v ⇒ T_t⁻u ≤ T_t⁻v.

(4) If c is a constant and u ∈ C⁰(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 ∈ C⁰(M, R), ∀t ≥ 0, kT_t⁻u − T_t⁻vk_∞≤ ku − vk∞. (6) For each u ∈ C⁰(M, R), we have lim_t→0T_t⁻u = u.

(7) For each u ∈ C⁰(M, R), the map t 7→ T_t⁻u is uniformly continuous.

(8) For each u ∈ C⁰(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 t₀> 0, the family of functions (t, x) 7→ T_t⁻u(x), u ∈ C⁰(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 C¹ maps form a dense subset of C⁰(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_Kt.

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

L(γ(s), ˙γ(s)) ds + u(γ(0)) ≥ u(γ(t)) + C_Kt, which gives

T_t⁻u(x) ≥ u(x) + C_Kt.

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 A₀ = max_x∈ML(x, 0), we found that kT_t⁻u − uk_∞≤ t max(C_KA₀),

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 t₀ > 0. By Lemma 4.4.2, we know that there exists a finite constant K(t₀) such that T_t⁻u is Lipschitz with Lipschitz constant

≤ K(t0), for each u ∈ C⁰(M, R) and each t ≥ t₀. It follows from the semi-group property and the proof of (6) that

kT_t⁻^′ u − T_t⁻uk_∞≤ (t^′− t) max(C_K(t₀₎, A₀),

for all t^′≥ t ≥ t0. It is then easy to check that (t, x) 7→ T_t⁻u(x) is Lipschitz on [t₀, +∞[×M , with Lipschitz constant ≤ max(C_K(t₀₎, A₀)+

K(t₀), 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_1/2ⁿ, 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 = C⁰(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 ∈ C⁰(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₋ ∈ C⁰(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 = −c₁. 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 ∈ C⁰(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_tu₋(x) = u₋(γ_t(0)) + Z t

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_tis (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)) ∈ K₁.

Since the ¯γ_tare 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 K₁, we can

141 find a sequence t_nր +∞ 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

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

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

T M

[u₋(πφ₁(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 K₁of 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 K₁. 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

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_xu₋) = c[0].

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

H(x, d_xu₋) = sup

v∈TxM

d_xu₋(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_xu₋( ˙γ₋^x(0)) = L(x, ˙γ₋^x(0)) + c[0].

Nel documento Weak KAM Theorem in Lagrangian Dynamics Seventh Preliminary Version (pagine 146-159)