A natural problem in dynamics is the search for subsets invari-ant by the flow φs. Within the framework which concerns us the Hamilton-Jacobi method makes it possible to find such invariant subsets.

To explain this method, it is better to think of the Hamiltonian
H as a function on cotangent bundle T^{∗}M . Indeed, under the
assumptions (1) and (2) above, we see that the Legendre transform
L : T M → T^{∗}M , defined by

L(x, v) = (x,∂L

∂v(x, v)),

is a diffeomorphism of T M onto T^{∗}M . We can then regard H as
a function on T^{∗}M defined by

H(x, p) = p(v) − L(x, v), where p = ∂L

∂v(x, v).

As the Legendre transform L is a diffeomorphism, we can use it
to transport the flow φ_{t}: T M → T M to a flow φ^{∗}_{t} : T^{∗}M → T^{∗}M
defined by φ^{∗}_{t} = Lφ_{t}L^{−1}.

Theorem 0.1.1 (Hamilton-Jacobi). Let ω be a closed 1-form on
M . If H is constant on the graph Graph(ω) = {(x, ω_{n}) | x ∈ M },
then this graph is invariant by φ^{∗}_{t}.

We can then ask the following question:

Given a fixed closed 1-form ω_{0}, does there exist ω another
closed 1-form cohomologous with ω_{0} such that H is constant on
the graph of ω?

The answer is in general negative if we require ω to be contin-uous. It is sometimes positive, this can be a way to formulate the Kolmogorov-Arnold-Moser theorem, see [Bos86].

However, there are always solutions in a generalized sense. In
order to explain this phenomenon, we will first show how to reduce
the problem to the 0 cohomology class. If ω_{0} is a fixed closed
1-form, let us consider the Lagrangian L_{ω}_{0} = L − ω_{0}, defined by

L_{ω}_{0}(x, v) = L(x, v) − ω_{0,x}(v).

Since ω_{0}is closed, if we consider only curves γ with the same fixed
endpoints, the map γ 7→ R

γω_{0} is locally constant. It follows that
L_{ω}_{0} and L have the same extremal curves. Thus they have also
the same Euler-Lagrange flow. The Hamiltonian H_{ω}_{0} associated
with L_{ω}_{0} verifies

H_{ω}_{0}(x, p) = H(x, ω_{0,x}+ p).

By changing the Lagrangian in this way we see that we have only
to consider the case ω_{0} = 0.

We can then try to solve the following problem:

Does there exist a constant c ∈ R and a differentiable function
u : M → R such that H(x, d_{x}u) = c, for all x ∈ M ?

There is an “integrated” version of this question using the
semi-group T_{t}^{−}: C^{0}(M, R) → C^{0}(M, R), defined for t ≥ 0 by

T_{t}^{−}u(x) = inf{L(γ) + u(γ(0)) | γ : [0, t] → M, γ(t) = x}.

It can be checked that T_{t+t}^{−} ′ = T_{t}^{−}◦ T_{t}^{−}′, and thus T_{t}^{−} is a
(non-linear) semigroup on C^{0}(M, R).

A C^{1} function u : M → R, and a constant c ∈ R satisfy
H(x, d_{x}u) = c, for all x ∈ M , if and only if T_{t}^{−}u = u − ct, for each
t ≥ 0.

Theorem 0.1.2 (Weak KAM). We can always find a Lipschitz
function u : M → R and a constant c ∈ R such that T_{t}^{−}u = u − ct,
for all t ≥ 0.

xiii
The case M = T^{n}, in a slightly different form (viscosity
solu-tions) is due to P.L. Lions, G. Papanicolaou and S.R.S.
Varadha-ran 87, see [LPV87, Theorem 1, page 6]. This general version was
obtained by the author in 96, see [Fat97b, Th´eor`eme1, page 1044].

Carlsson, Haurie and Leizarowitz also obtained a version of this theorem in 1992, see [CHL91, Theorem 5.9, page 115].

As u is a Lipschitz function, it is differentiable almost
every-where by Rademacher’s Theorem. It can be shown that H(x, d_{x}u) =
c at each point where u is differentiable. Moreover, for such a
func-tion u we can find, for each x ∈ M , a C^{1} curve γx:] − ∞, 0] → M ,
with γ_{x}(0) = x, which is a solution of the multivalued vector field

“ grad_{L}u”(x) defined on M by

“ grad_{L}u”(y) = L^{−1}(y, d_{y}u).

These trajectories of grad_{L}u are minimizing extremal curves.

The accumulation points of their speed curves in T M for t → −∞

define a compact subset of T M invariant under the Euler-Lagrange
flow ϕ_{t}. This is an instance of the so-called Aubry and Mather
sets found for twist maps independently by Aubry and Mather in
1982 and in this full generality by Mather in 1988.

We can of course vary the cohomology class replacing L by
L_{ω} and thus obtain other extremal curves whose speed curves
define compact sets in T M invariant under φ_{t}. The study of these
extremal curves is important for the understanding of this type of
Lagrangian Dynamical Systems.

### Chapter 1

## Convex Functions:

## Legendre and Fenchel

Besides some generalities in the first two sections, our main goal in this chapter is to look at the Legendre and Fenchel transforms.

This is now standard material in Convex Analysis, Optimization, and Calculus of Variations. We have departed from the usual viewpoint in Convex Analysis by not allowing our convex functions to take the value +∞. We think that this helps to grasp things on a first introduction; moreover, in our applications all functions have finite values. In the same way, we have not considered lower semi-continuous functions, since we will be mainly working with convex functions on finite dimensional spaces.

We will suppose known the theory of convex functions of one real variable, see for example [RV73, Chapter 1]or [Bou76, Chapitre 1].

### 1.1 Convex Functions: General Facts

Definition 1.1.1 (Convex Function). Let U be a convex set in the vector space E. A function f : U → R is said to be convex if it satisfies the following condition

∀x, y ∈ U, ∀t ∈ [0, 1], f (tx + (1 − t)y) ≤ tf (x) + (1 − t)f (y).

1

The function f is said to be strictly convex if it satisfies the fol-lowing stronger condition

∀x 6= y ∈ U, ∀t ∈]0, 1[, f (tx + (1 − t)y) < tf (x) + (1 − t)f (y).

It results from the definition that f : U → R is convex if and only if for every line D ⊂ E the restriction of f on D ∩ U is a convex function (of one real variable).

Proposition 1.1.2. (i) An affine function is convex (an affine function is a function which is the sum of a linear function and a constant).

(ii) If (f_{i})_{i} ∈ I is a family of convex functions : U → R, and
sup_{i∈I}f_{i}(x) < +∞ for each x ∈ U then sup_{i∈I}f_{i} is convex.

(iii) Let U be an open convex subset of the normed space E.

If f : U → R is convex and twice differentiable at x ∈ U , then
D^{2}f (x) is non-negative definite as a quadratic form. Conversely,
if g : U → R admits a second derivative D^{2}g(x) at every point
x ∈ U , with D^{2}g(x) non-negative (resp. positive) definite as a
quadratic form, then g is (resp. strictly) convex.

Properties (i) and (ii) are immediate from the definitions. The property (iii) results from the case of the functions of a real vari-able by looking at the restrictions to each line of E.

Definition 1.1.3 (C^{2}-Strictly Convex Function). Let be a in the
vector space E. A function f : U → R, defined on the convex
subset U of the normed vector space E, is said to be C^{2}-strictly
convex if it is C^{2}, and its the second derivative D^{2}f (x) is positive
definite as a quadratic form, for each x ∈ U .

Exercise 1.1.4. Let U be an open convex subset of the normed space E, and let f : U → R be a convex function.

a) Show that f is not strictly convex if and only if there exists a pair of distinct points x, y ∈ U such that f is affine on the segment [x, y] = {tx + (1 − t)y | t ∈ [0, 1].

b) If f is twice differentiable at every x ∈ U , show that it
is strictly convex if and only if for every unit vector v ∈ E the
set {x ∈ U | D^{2}f (x)(v, v) = 0} does not contain a non trivial
segment parallel to v. In particular, if D^{2}f (x) is non-negative
definite as a quadratic form at every point x ∈ U , and the set of

3
points x where D^{2}f (x) is not of maximal rank does not contain a
non-trivial segment then f is strictly convex.

Theorem 1.1.5. Suppose that U is an open convex subset of
the topological vector space E. Let f : U → R be a convex
function. If there exists an open non-empty subset V ⊂ U with
sup_{x∈V} f (x) < +∞, then f is continuous on U.

Proof. Let us first show that for all x ∈ U , there exists an open
neighborhood V_{x} of x, with V_{x} ⊂ U and sup_{y∈V}_{x}f (y) < +∞.
In-deed, if x /∈ V , we choose z0 ∈ V . The intersection of the open set
U and the line containing x and z0 is an open segment
contain-ing the compact segment [x, z_{0}]. We choose in this intersection, a
point y near to x and such that y /∈ [x, z0], thus x ∈]y, z_{0}[, see
fig-ure 1.1. It follows that there exists t with 0 < t0 < 1 and such that
x = t_{0}y + (1 − t_{0})z_{0}. The map H : E → E, z 7→ x = t_{0}y + (1 − t_{0})z
sends z_{0}to x, is a homeomorphism of E, and, by convexity of U , it
maps U into itself. The image of V by H is an open neighborhood
V_{x} of x contained in U . Observe now that any point x^{′} of V_{x} can
be written as the form x^{′} = t_{0}y + (1 − t_{0})z with z ∈ V for the
same t_{0} ∈]0, 1[ as above, thus

f (x^{′}) = f (t_{0}y + (1 − t_{0})z)

≤ t0f (y) + (1 − t_{0})f (z)

≤ t_{0}f (y) + (1 − t_{0}) sup

z∈V

f (z) < +∞.

This proves that f is bounded above on V_{x}.

Let us now show that f is continuous at x ∈ U . We can
suppose by translation that x = 0. Let V_{0} be an open subset of U
containing 0 and such that sup_{y∈V}_{0}f (y) = M < +∞. Since E is
a topological vector space, we can find an open set ˜V_{0} containing
0, and such that t ˜V_{0} ⊂ V_{0}, for all t ∈ R with |t| ≤ 1. Let us
suppose that y ∈ ǫ ˜V_{0}∩ (−ǫ ˜V_{0}), with ǫ ≤ 1. We can write y = ǫz_{+}
and y = −ǫz−, with z+, z− ∈ ˜V0 (of course z− = −z+, but this
is irrelevant in our argument). As y = (1 − ǫ)0 + ǫz_{+}, we obtain
f (y) ≤ (1 − ǫ)f (0) + ǫf (z_{+}), hence

∀y ∈ ǫ ˜V_{0}∩ (−ǫ ˜V_{0}), f (y) − f (0) ≤ ǫ(M − f (0)).

U

V
V_{x}

z_{0}

x y

Figure 1.1:

We can also write 0 = _{1+ǫ}^{1} y + _{1+ǫ}^{ǫ} z_{−}, hence f (0) ≤ _{1+ǫ}^{1} f (y) +

ǫ

1+ǫf (z_{−}) which gives (1 + ǫ)f (0) ≤ f (y) + ǫf (z_{−}) ≤ f (y) + ǫM .
Consequently

∀y ∈ ǫ ˜V_{0}∩ (−ǫ ˜V_{0}), f (y) − f (0) ≥ −ǫM + ǫf (0).

Gathering the two inequalities we obtain

∀y ∈ ǫ ˜V_{0}∩ (−ǫ ˜V_{0}), |f (y) − f (0)| ≤ ǫ(M − f (0)).

Corollary 1.1.6. A convex function f : U → R defined on an
open convex subset U of R^{n}is continuous.

Proof. Let us consider n+1 affinely independent points x_{0}, · · · , x_{n}∈
U . The convex hull σ of x_{0}, · · · , x_{n} has a non-empty interior. By
convexity, the map f is bounded by max^{n}_{i=0}f (x_{i}) on σ.

Most books treating convex functions from the point of view of Convex Analysis do emphasize the role of lower semi-continuous convex functions. When dealing with finite valued functions, the following exercise shows that this is not really necessary.

5 Exercise 1.1.7. Let U be an open subset of the Banach space E.

If f : U → R is convex, and lower semi-continuous show that it is
in fact continuous. [Indication: Consider the sequence of subsets
C_{n} = {x ∈ U | f (x) ≤ n}, n ∈ N. Show that one of these subsets
has non-empty interior.]

We recall that a function f : X → Y , between the metric
spaces X, Y , is said to be locally Lipschitz if, for each x ∈ X,
there exists a neighborhood Vx of x in X on which the restriction
f_{|V}_{x} is Lipschitz.

Theorem 1.1.8. Let E be a normed space and U ⊂ E an open convex subset. Any convex continuous function f : U → R is a locally Lipschitz function.

Proof. In fact, this follows from the end of the proof of Theorem 1.1.5. We now give a direct slightly modified proof.

We fix x ∈ U . Since f is continuous, there exists r ∈]0, +∞[

and M < +∞ such that sup

y∈ ¯B(x,r)

|f (y)| ≤ M.

We have used the usual notation ¯B(x, r) to mean the closed ball of center x and radius r.

Let us fix y, y^{′} ∈ ¯B(x, r/2). We call z the intersection point
of the boundary ∂B(x, r) = {x^{′} ∈ E | kx^{′}− xk = r} of the closed
ball ¯B(x, r) with the line connecting y and y^{′} such that y is in the
segment [z, y^{′}], see figure 1.2. We of course have kz−y^{′}k ≥ r/2. We
write y = tz + (1 − t)y^{′}, with t ∈ [0, 1[, from which it follows that
y−y^{′}= t(z−y^{′}). By taking the norms and by using kz−y^{′}k ≥ r/2,
we see that

t ≤ ky^{′}− yk2
r.

The convexity of f gives us f (y) ≤ tf (z)+ (1− t)f (y^{′}), from which
we obtain the inequality f (y) − f (y^{′}) ≤ t(f (z) − f (y^{′})). It results
that

f (y) − f (y^{′}) ≤ 2tM ≤ 4M

r ky − y^{′}k,
and by symmetry

∀y, y^{′} ∈ ¯B(x, r/2), |f (y) − f (y^{′})| ≤ 4M

r ky − y^{′}k.

z

y

y^{′}

Figure 1.2:

Corollary 1.1.9. If f : U → R is convex with U ⊂ R^{n} open and
convex, then f is a locally Lipschitz function.

We recall Rademacher’s Theorem, see [EG92, Theorem 2, page 81]or [Smi83, Theorem 5.1, page 388].

Theorem 1.1.10 (Rademacher). A locally Lipschitz function
de-fined on open subset of R^{n} and with values in R^{m} is Lebesgue
almost everywhere differentiable.

Corollary 1.1.11. A convex function f : U → R, where U is
open convex of R^{n}, is Lebesgue almost everywhere differentiable.

It is possible to give a proof of this Corollary not relying on Rademacher’s Theorem, see [RV73, Theorem D, page 116]. We conclude this section with a very useful lemma.

Lemma 1.1.12. Let f : V → R be a convex function defined on an open subset V of a topological vector space.

(a) A local minimum for f is a global minimum.

7 (b) If f is strictly convex, then f admits at most one minimum.

Proof. (a) Let x_{0} be a local minimum. For y ∈ U and t ∈ [0, 1]

and close to 1 we have

f (x_{0}) ≤ f (tx_{0}+ (1 − t)y) ≤ tf (x_{0}) + (1 − t)f (y),

thus (1 − t)f (x0) ≤ (1 − t)f (y) for t close to 1. It follows that
f (y) ≥ f (x_{0}).

(b) It results from the convexity of f that the subset {x | f (x) ≤ λ} is convex. If λ = inf f , we have {x | f (x) = inf f } = {x | f (x) ≤ inf f }. If f is strictly convex this convex set cannot contain more than one point.