Maxime Zavidovique

Discrete and Continuous Weak KAM Theory:
an introduction through examples
and its applications to twist maps.

###### Abstract

The aim of these notes is to present a self contained account of discrete weak
KAM theory. Put aside the intrinsic elegance of this theory, it is also a toy
model for classical weak KAM theory, where many technical difficulties
disappear, but where central ideas and results persist. It can therefore serve
as a good introduction to (continuous) weak KAM theory. After a general
exposition of the general abstract theory, several examples are studied. The
last section is devoted to the historical problem of conservative twist maps
of the annulus. At the end of the first three Chapters, the relations between
the results proved in the discrete setting and the analogous theorems of
classical weak KAM theory are discussed. Some key differences are also
highlighted between the discrete and classical theory.

À mes trois rayons de soleil,
du matin, du midi et du soir.

## Remerciements

Je voudrais avant tout remercier Thétis. C’est probablement grâce à sa
négligence que j’ai eu le temps de me remettre à la rédaction de ce mémoire,
confortablement immobilisé dans mon canapé.

Un grand merci également à Symplectix et à ses organisateurs de m’accueillir
subrepticement le temps d’une soutenance.

As far as real living persons are concerned, my gratitude goes to Alessio
Figalli, Hitoshi Ishii and Claude Viterbo for sacrificing part of their summer
time to the task of reading my prose, thank you.

Merci à Alessio, Claude, Marie-Claude Arnaud, Pierre Cardaliaguet, Albert
Fathi, Hélène Frankowska, Vadim Kaloshin et Ludovic Rifford d’avoir accepté
d’être dans le jury de cette HDR. Votre présence m’honore.

Je suis à jamais redevable à Albert de m’avoir introduit dans le monde KAM
faible, il y a maintenant quinze ans. C’est un monde d’idées simples et
profondes, un monde parfois technique mais rarement trop. L’explorer est une
jubilation.

Questo testo è nato dopo un mini corso dato a Cortona en 2015. Inoltre di
mangiare tartufo con Montelpuciano, mi ha fatto pensare a come organizzare la
teoria di maniera pedagogica. Grazie agli organizzatori Andrea, Antonio e
Alfonso per questa stupende conferenza.

Ce texte, et une partie de son contenu, n’aurait pas été écrit sans mes
coauteurs, en particulier, Andrea, Albert, Antonio et bien sûr Marie-Claude.
Apprendre grâce à vous et découvrir ensemble de nouveaux théorèmes suscite
chez moi émerveillement et plaisir. Les mathématiques sont un sport d’équipe !

Parlant d’équipe, merci à Nicolas Bergeron, Alexandru Oancea et Patrice Le
Calvez pour votre accompagnement et soutien à Jussieu. Alex, merci aussi et
surtout pour tes précieux conseils et la relecture de mon texte. Évoluer dans
un environment professionnel chaleureux et apaisé est un luxe111que SU facture
380€ dont j’ai la chance de bénéficier. C’est grâce à tous mes collègues de
AA. Une spéciale dédicace à Antonin et Pierre, merci pour votre aide précieuse
avec les figures ! Je n’oublie pas les anciens collègues, Hélène, Erwan et
surtout Grégory pour la formation continue en chocolat et pâtisserie, et pour
tout le reste.

Cette soutenance n’aurait pas existé (et moi non plus) sans mes parents. Merci
pour vos encouragements pendant la longue phase de rédaction. Ils furent
réguliers, soutenus mais toujours avec une pudique retenue. Jamais plus de
trois fois par jour.

Mes frères et sœurs m’accompagnent tout le temps, je pense à eux. Tout
particulièrement à Laurent, merci pour le grand flacon.

Enfin, et surtout, Anne et les enfants illuminent mon quotidien. Si les
enfants sont mon soleil, Anne tu es mon Étoile polaire.

###### Contents

  1. Introduction
    1. Weak KAM, a bridge between Aubry–Mather and Viscosity solutions
      1. Aubry–Mather theory
      2. Viscosity Solutions
      3. The bridge
    2. Weak KAM, beyond Hamilton–Jacobi equations
      1. Optimal control theory
      2. Contact type and systems of Hamilton–Jacobi equations
      3. Lorentzian geometry and Lyapunov functions
      4. Optimal Transportation
      5. Discrete weak KAM theory, with an economical twist
    3. Organisation of the text
  2. 1 The discrete setting, weak KAM solutions and subsolutions
    1. 1.1 Discrete setting and the Lax–Oleinik semigroup
    2. 1.2 The weak KAM Theorem and critical subsolutions
    3. 1.3 The positive Lax–Oleinik semigroup
    4. 1.4 Strict subsolutions, Aubry sets
    5. 1.5 Relations to the classical theory
      1. 1.5.1 Classical setting and Lax-Oleinik semigroup
      2. 1.5.2 The weak KAM Theorem and critical subsolutions
      3. 1.5.3 The positive Lax–Oleinik semigroup
      4. 1.5.4 Strict subsolutions, Aubry sets
  3. 2 More (dynamical) characterizations of the Aubry sets
    1. 2.1 The Peierls Barrier
    2. 2.2 Examples of points in the Aubry sets
    3. 2.3 Regularity of subsolutions
    4. 2.4 More regularity for subsolutions
    5. 2.5 Graph properties and dynamics on the Aubry set
    6. 2.6 Relations to the classical theory
      1. 2.6.1 The classical Peierls Barrier
      2. 2.6.2 Examples of points in the Aubry set
      3. 2.6.3 Regularity and more regularity of subsolutions
      4. 2.6.4 Graph properties, twist condition and dynamics on the Aubry set
  4. 3 Minimizing Mather measures and the discounted semigroups
    1. 3.1 Minimizing Mather measures
      1. 3.1.1 An optimal transport like approach
      2. 3.1.2 An ergodic point of view
    2. 3.2 The discounted equation
    3. 3.3 Discount for the positive Lax–Oleinik semigroup
    4. 3.4 Degenerate discounted equations
    5. 3.5 Comment on the discounted procedure
    6. 3.6 Relations to the classical theory
      1. 3.6.1 Minimizing Mather measures
      2. 3.6.2 The classical discounted equation
      3. 3.6.3 Discount for the positive classical L.–O. semigroup
      4. 3.6.4 Some degenerate discounted Hamilton–Jacobi equations
  5. 4 A family of examples
    1. 4.1 The study of $H_{0}$
    2. 4.2 Increasing the cohomology class: $\boxed{c\in[0,\alpha]}$
    3. 4.3 A change of regime: $\boxed{c\in\big{(}\alpha,\int_{0}^{1}f^{+}(x){\rm d}x\big{)}}$
    4. 4.4 The limiting case: $\boxed{c_{0}=\int_{0}^{1}f^{+}(x){\rm d}x}$
    5. 4.5 Positive rotation numbers: $\boxed{c>\int_{0}^{1}f^{+}(x){\rm d}x}$
      1. 4.5.1 Non–continuity of $u_{1}^{c}$ with respect to $c$
      2. 4.5.2 A situation where $u_{1}^{c}\neq U_{0}^{c}$
    6. 4.6 Concluding example
  6. 5 Twist maps
    1. 5.1 Definitions and variational structure
      1. 5.1.1 Definition and Birkhoff’s theorem
      2. 5.1.2 The generating function, properties and consequences
    2. 5.2 Examples and Moser’s Theorem
      1. 5.2.1 Notions of integrability
      2. 5.2.2 The standard family
      3. 5.2.3 General twist maps and a Theorem of Moser
    3. 5.3 Weak KAM for twist maps
    4. 5.4 Mather measures
    5. 5.5 Order properties of weak KAM solutions
    6. 5.6 Structure of infinite minimizing chains
      1. 5.6.1 Irrational rotation number
      2. 5.6.2 Rational rotation number
    7. 5.7 A glimpse into the world of weakly integrable twist maps
  7. Bibliography

## Introduction

The present text initially emerged from lecture notes of a course given in
Cortona in 2015 at the INDAM meeting entitled The Hamilton–Jacobi Equation: At
the crossroads of PDE, dynamical systems & geometry. The goal of the lectures
was to give a complete and thorough introduction to weak KAM theory through
the prism of its discrete pendant. It culminated with the proof of convergence
of the solutions of the discounted equation, which was new at the time.

The pedagogical motivation is that discrete weak KAM theory is peculiarly
elementary. Basic topology is the only prerequisite and the most advanced
tools are the Arzelà–Ascoli Theorem and weak compactness of probability
measures on a compact metric space. However, all important results of weak KAM
theory find their analogue in discrete weak KAM theory and the proofs being
stripped of technicalities reveal the key ideas more clearly. After studying
this toy model, the interested reader can then go on to learn more on the
major theories, as Calculus of Variations, Viscosity solutions of
Hamilton–Jacobi equations, and Tonelli Hamiltonian Dynamical Systems. The
latter are at the core of classical weak KAM theory.

We wish to start by explaining in which context weak KAM theory emerged in the
90’s. Most notions described will be rigorously defined later in the body of
the text.

### Weak KAM, a bridge between Aubry–Mather and Viscosity solutions

Originally discovered by Albert Fathi in the 90’s [89, 88, 90, 91], weak KAM
theory was designed to understand the dynamical objects of Aubry–Mather Theory
for Tonelli Hamiltonian/Lagrangian systems through particular functions called
critical subsolutions and weak KAM solutions. As explained by Fathi, the term
weak KAM was chosen as KAM tori give rise to strong solutions of the
Hamilton–Jacobi equation while weak KAM solutions are weak solutions of the
same equation. It turned out that weak KAM solutions and subsolutions fall in
the realm of Viscosity Solutions and subsolutions of Hamilton–Jacobi
equations, a theory founded by Crandall and Lions [74].

#### Aubry–Mather theory

This theory originated in the study of conservative twist maps and
Frenkel–Kontorova models. The objective was to understand invariant sets and
the structure of minimizing orbits of such systems. Another type of problem
was to construct diffusion orbits connecting invariant sets, or to understand
obstructions for such orbits to exist. Aubry & Le Daeron [18] and Mather [138]
understood that orbits and invariant sets minimizing a certain energy verify
similar properties as orbits of homeomorphisms of the circle. Hence they could
apply Poincaré-Denjoy theory to classify and understand such invariant sets
according to their rotation number. They went on to study the minimal average
action of such minimizing orbits (a function only depending on the rotation
number), invariant minimal measures (that would become Mather measures) and
Mather gave several definitive answers to questions on the existence of
connecting orbits [140, 141, 139, 142, 144]. Amongst other important
contributions let us mention Bangert [21, 22].

Motivated by Moser, Mather developed a generalization of his theory to higher
dimensional settings by introducing Minimizing Measures for a Tonelli
Lagrangian defined on the tangent bundle of a compact manifold [143]. His next
goal would then be to use this tool to tackle Arnol’d diffusion of such
systems, a phenomenon highlighted by Arnol’d in his famous examples [17]. Let
us present the philosophy. In an integrable system, all orbits are bounded and
periodic or quasi–periodic. If one perturbs such a system, KAM theory implies
that many quasi–periodic orbits persist. However, Arnol’d constructed examples
where for small perturbations, some orbits have a huge drift in energy. He
then conjectured that such a phenomenon should be typical. Mather led the way
proposing groundbreaking variational mechanisms to construct generic diffusion
([145, 146, 147]). Since then, there has been a huge literature trying to
carry on Mather’s program for diffusion.

#### Viscosity Solutions

As already mentioned, viscosity solutions were introduced by Crandall and
Lions. They provide a simple, robust definition for weak solutions of first
and second order PDE’s. Strong existence and uniqueness results are obtained
for wide classes of equations, including Hamilton–Jacobi equations (stationary
and evolutionary) making the solutions worthwhile studying. Let us mention
amongst many others the founding works of Ishii, Crandall, Lions [115, 72,
121, 116]. The definition, that makes use of test functions that are
super–tangent or sub–tangent to the solution is very geometric, and allows
easily to obtain stability results for viscosity solutions. References to
learn more about basic (and more advanced) properties are [131, 73, 24]. For
example, viscosity solution theory is so flexible as to apply to
non–continuous functions. Stability allows to obtain very general convergence
theorems of approximation schemes [164, 25]. Other references making use of
this idea, in more weak KAM or variational contexts, are [51, 176, 160, 172].

Very soon in the development of the theory of Viscosity Solutions, Lions and
collaborators realized the important role of dynamical programming properties
for evolutionary equations and the links with Optimal Control theory that
naturally emerge. Indeed, the Value function in optimal control is almost
systematically a viscosity solution to some Hamilton–Jacobi equation [23].
This is fundamental in weak KAM theory as the Lax–Oleinik semigroup
rediscovered by Fathi turns out to be the value function of an optimal control
problem.

Another important problem that was solved early on by viscosity solution
methods is that of Homogenization of the Hamilton–Jacobi equation. Consider a
continuous, $\mathbb{Z}^{N}$–periodic in the first variable Hamiltonian
$H:\mathbb{R}^{N}\times\mathbb{R}^{N}$ that is uniformly coercive in the
second variable. Fix a bounded and uniformly continuous initial data
$u_{0}:\mathbb{R}^{N}\to\mathbb{R}$. Given $\varepsilon>0$, Lions,
Papanicolaou and Varadhan consider the evolutionary Hamilton–Jacobi equation

$\begin{cases}\partial_{t}U+H(\frac{x}{\varepsilon},\partial_{x}U)=0,\\\
U(0,\cdot)=u_{0},\end{cases}$ (EHJ$\varepsilon$)

that admits a unique viscosity solution
$U_{\varepsilon}:[0,+\infty)\times\mathbb{R}^{N}\to\mathbb{R}$. They prove
that as $\varepsilon\to 0$, the functions $U_{\varepsilon}$ converge locally
uniformly to a function $U_{0}$ that is characterized as the unique solution
to

$\begin{cases}\partial_{t}U+\overline{H}(\partial_{x}U)=0,\\\
U(0,\cdot)=u_{0},\end{cases}$ (E$\overline{H}$J)

where $\overline{H}$ is called the effective Hamiltonian. It is defined as
follows: for $P\in\mathbb{R}^{N}$, $\overline{H}(P)$ is the only constant such
that the cell problem $H(x,P+\partial_{x}u)=\overline{H}(P)$ admits a
$\mathbb{Z}^{N}$–periodic viscosity solution $u:\mathbb{R}^{N}\to\mathbb{R}$.
In Homogenization theory, the state variable $x\in\mathbb{R}^{N}$ takes values
in the universal cover of the flat torus $\mathbb{T}^{N}$. A fundamental
domain of the covering map is then the cell $[0,1)^{N}$ in the sense that the
knowledge of a $\mathbb{Z}^{N}$–periodic function on $\mathbb{R}^{N}$ is
equivalent to its restriction to the cell $[0,1)^{N}$. The name cell problem
comes from the fact that the unknown function is defined on a cell.

This effective Hamiltonian (in the case of convex Hamiltonians) coincides with
the minimal average action, Mather’s $\alpha$ function. The solutions to the
cell problem in our terminology will be weak KAM solutions. This problem was
revisited by Evans in [85] where he introduced the perturbed test function
method. Much later, it was studied under the light of symplectic topology by
Viterbo and Montzner, Vichery, Zapolsky [166, 152]. Amongst many other follow
ups in the spirit of Lions Papanicolaou and Varadhan, let us mention recent
generalizations to other manifolds [71, 163].

#### The bridge

As all good bridges, weak KAM theory quickly helped the development of both
banks it joins. From a dynamical point of view, Fathi’s first achievement was
to construct connecting orbits through conjugate pairs of weak KAM solutions
[88, 90].

From the PDE side, he proved long time convergence of solutions to the
evolutionary Hamilon–Jacobi equation on compact manifolds [91], for autonomous
Hamiltonians. Though partial results had been obtained by PDE means [154], the
new idea he imported from the dynamical world is that long minimizing
trajectories tend to accumulate on the support of minimizing Mather measures.
This was followed by many generalizations, for instance [77, 26, 117] where
variational and PDE methods allow to weaken regularity hypotheses that
Dynamical methods require. In a sense, this culminates in works (the first one
being [53]) which use an idea of Evans [86] where Mather measures are given a
PDE definition and henceforth adapted to more general settings. Other related
results are :

  * •

for counterexamples in the non–autonomous setting [96, 27];

  * •

for positive results in dimension $1$ [32, 43].

Another question raised in the theory of Hamilton–Jacobi equations was that of
regularity of critical subsolutions. For Tonelli Hamiltonians, this was
settled by Fathi and Siconolfi in [99] where existence of $C^{1}$ subsolutions
is proved. Along the way, the authors also gave a simple proof of Mañé’s
characterizations of Mather measures, as closed minimizing measures ([133,
134]). This is the point of view used in this work. In [100] the same authors
extended their results to Lipschitz Hamiltonians. Finally, in [33], existence
of $C^{1,1}$ subsolutions is obtained by some Lasry–Lions type approximation
([129, 36]). This could be expected as Fathi had also proved that $C^{1}$
solutions are automatically $C^{1,1}$ ([92]).

The latest breakthrough of weak KAM theory in the PDE theory of
Hamilton–Jacobi equations is probably the proof of convergence of the
solutions of the discounted equations [75], following some special cases in
[125]. The discounting method was an approximation procedure used already in
[132] to prove existence of solutions to the cell problem. It has the
advantage to approximate it by equations verifying a strong comparison
principle and having exactly one solution. The convergence result of Davini,
Fathi, Iturriaga and the author is that, as the perturbation goes to $0$, a
particular weak KAM solution is selected when the Hamiltonian is convex in the
second variable. Prior conditions on selected limiting solutions had been
found by Gomes in [109]. The result strikes by its generality (little
regularity is assumed on the Hamiltonian, no strict convexity) and by the
flexibility of its proof. It was naturally followed by numerous
generalizations and adaptations. Among them let us mention:

  * •

the discrete setting that is presented later in this text [75],

  * •

adaptations to Neumann problems [1],

  * •

non–compact settings [124],

  * •

more abstract duality methods [122, 123],

  * •

second order Hamilton–Jacobi equations [150],

  * •

weakly coupled systems of Hamilton–Jacobi equations [82] using weak KAM tools
from [78] and [119, 120] for more general results,

  * •

convergence from the negative direction [79],

  * •

for more general nonlinear discount approximations [64, 171, 63, 65],

  * •

for discounted approximations on networks instead of manifolds [157],

  * •

for mean field games [61]

  * •

more recently, with degenerate discounting approximations [177].

Some limitations and counterexamples exist nevertheless when convexity is
dropped [180, 118] and last but not least, let us quote [13] for a more
geometric perspective on the convergence result.

Crossing back to the dynamical systems world, Patrick Bernard used weak KAM
solutions to push Mather’s ideas in [32, 34]. Probably the most definitive
works on Arnol’d diffusion to this day are [35, 42]. Those works mix on the
one hand dynamical strategies dating back to Arnol’d following chains of
normally hyperbolic objects having transverse intersections of stable and
unstable manifolds, and on the other hand analytical tools about regularity of
weak KAM solutions and subsolutions.

In his founding works [133, 134], Mañé proved that a generic Hamiltonian (in a
sense he defined), has a unique Mather measure that is hence ergodic for the
Lagrangian flow. He then asked if this measure is generically concentrated on
a hyperbolic periodic orbit. This question is known as Mañé’s conjecture. It
is still open but important steps were made by Figalli and Rifford [103, 104]
who managed to bring together methods of optimal control and PDE with the
dynamical system theory of orbit closing lemma. They prove the conjecture in
low regularity assuming the existence of a smooth critical subsolution. Then
with Contreras [70] they brought in Arnaud’s theory of Green bundles [4] to
prove that generically, the Aubry set is hyperbolic if the underlying manifold
has dimension 2. The idea is that in this setting, if the Aubry set is not
hyperbolic, then positive and negative Green bundles coincide and weak KAM
solutions gain some extra regularity allowing to use ideas from the previous
works.

Mather had also raised a similar question about obtaining a uniform bound on
the number of ergodic Mather measures when the cohomology varies, for generic
Hamiltonians. This was solved by Bernard and Contreras [41, 37]. More
generally, understanding the shape of the Aubry set is a challenging question
that is still to be understood. Progress on the quotiented Aubry set was
obtained by Sorrentino and Fathi, Figalli, Rifford in [162, 95] and for the
actual Aubry set, by Arnaud [10].

### Weak KAM, beyond Hamilton–Jacobi equations

It turns out that the philosophy of weak KAM theory and of Aubry–Mather theory
applies to a variety of other areas.

#### Optimal control theory

Weak KAM theory’s starting point is the fact that solutions to Hamilton–Jacobi
equations of the form (EHJ$\varepsilon$) (seen in the context of
homogenization), on a manifold $M$, when the Hamiltonian has some convexity
properties, are given by an explicit formula called Lax–Oleinik semigroup (for
$\varepsilon=1$):

$U(t,x)=\inf_{\begin{subarray}{c}\gamma:[-t,0]\to M\\\
\gamma(0)=x\end{subarray}}u_{0}\big{(}\gamma(-t)\big{)}+\int_{-t}^{0}L\big{(}\gamma(s),\dot{\gamma}(s)\big{)}{\rm
d}s,$ (1)

where the infimum is taken amongst all absolutely continuous curves. This
formula (or its variations) is a convolution in the (min,+) semiring and is
widely studied in Optimal Control theory. In this setting, $U$ is called the
value function. Excellent introductory references on the subject are [60, 23,
68] and we already mentioned how Fathi and others made groundbreaking progress
by importing ideas from Optimal Control (such as semiconcavity, regularity
properties of minimizers…). Contributions in the other directions are also
worth highlighting. Recently for instance, conjectures about propagation of
singularities of solutions of Hamilton–Jacobi equations were solved by the use
of the positive Lax–Oleinik semigroup [56, 57]. Such results find amazing
consequences in Riemannian geometry when applied to singularities of distance
functions [58].

#### Contact type and systems of Hamilton–Jacobi equations

Generalizations of weak KAM theory concern more general types of equations.
The first family of generalizations we have in mind is that of contact type
equations. This means that the Hamiltonian function
$H:T^{*}M\times\mathbb{R}\to\mathbb{R}$ also depends on the value $u(t,x)$ of
the unknown function. The terminology comes from the equations of
characteristics that preserve a contact form (instead of the symplectic form
for classical Hamiltonian equations). In this context, solutions are given by
an implicit Lax–Oleinik semigroup and weak KAM arguments can therefore be
used. This was exploited in many recent works including [168, 170, 59, 179,
64, 169, 65].

In a maybe more surprising way, weak KAM ideas also apply to some weakly
coupled systems of Hamilton–Jacobi equations. This is more unexpected as weak
KAM makes strong use of the order structure of $\mathbb{R}$ that is less
clearly adaptable for systems where the values taken are in $\mathbb{R}^{d}$
for some $d>1$. The first evidence of such a link was present in [54, 55] and
further developed in [80, 149, 78, 161, 113].

#### Lorentzian geometry and Lyapunov functions

As can be observed already from the definition of the Lax–Oleinik semigroup
(1) weak KAM theory studies minimization problems, objects verifying a family
of inequalities and minimizers of those inequalities. As a matter of fact,
central objects in weak KAM theory are critical subsolutions. They are
functions $u_{0}:M\to\mathbb{R}$ for which the associated function
$U(t,x)-\alpha(0)t$ (where $U$ is given by the Lax–Oleinik semigroup (1) and
$\alpha(0)$ is called the critical constant) is non–decreasing in $t$. Such
functions are also characterized by the following two properties:

  * •

$u_{0}$ is Lipschitz continuous,

  * •

$(x,D_{x}u_{0})\in H^{-1}(-\infty,\alpha(0)]\cap\\{x\\}\times T_{x}^{*}M$ for
almost every $x\in M$.

When $H$ is convex in the second variable, the sets
$H^{-1}(-\infty,\alpha(0)]\cap\\{x\\}\times T_{x}^{*}M$ are convex for all
$x$. Therefore a natural generalization is to replace the Hamiltonian by a
family of convex sets $C_{x}\in T^{*}_{x}M$ that verify suitable regularity
properties. The question being to know if solutions exist, one studies
differentiable inclusions of the form

  * •

$u$ is Lipschitz continuous,

  * •

$D_{x}u\in C_{x}$ for almost every $x\in M$.

In such settings, objects from Aubry–Mather theory such as the Aubry set
appear as obstructions to finding such functions that verify extra conditions
(as smoothness or replacing $C_{x}$ by its interior). Moreover, when possible,
weak KAM methods make it possible to construct $C^{1}$ solutions to such
differentiable inclusions. This was noticed by Fathi and Siconolfi [101] who
applied this philosophy to Lorentzian geometry, where the sets $C_{x}$ are
cones provided by a section of non positive $2$–forms. This line of research
was since developed in [44, 45, 165].

Another fruitful extension of weak KAM ideas concerns Lyapunov functions and
is closer to our subject of discrete weak KAM theory. Indeed, given a
continuous transformation $F$ of a metric space $X$, a Lyapunov function is
$f:X\to\mathbb{R}$ that is non–increasing (or when possible decreasing) on the
orbits of $F$. That is $f$ verifies the family of inequalities

$\forall x\in X,\quad f\circ F(x)\leqslant f(x).$

Clearly, no function can be decreasing on a periodic orbit. More generally any
reasonable notion of recurrence will provide an obstruction to the existence
of strict Lyapunov functions. Hence being able to construct optimal Lyapunov
functions is an important challenge related to fine dynamical properties. The
study of this problem, importing weak KAM ideas, was done in Pageault’s PhD
that led to recovering earlier results of Conley, Akin, Auslander… in simpler
and more precise form. Results are to be found in [155, 97, 98] and were
followed by further studies [48, 47, 49].

Let us also mention links with ergodic optimization and the analogue of Mañé’s
conjecture that was recently solved by Contreras [69]. Other works related to
weak KAM theory are [105, 106].

#### Optimal Transportation

Let us describe the original Monge problem in optimal transportation [151].
The goal is to move some material described by a probability measure $\mu$ on
a space $X$ into a certain configuration described by another probability
measure $\nu$ on a space $Y$, by means of a transport map $T:X\to Y$, knowing
that the cost to move material from $x\in X$ to $y\in Y$ is given by $c(x,y)$.
The problem is therefore to minimize $\int_{X}c\big{(}x,T(x)\big{)}\ {\rm
d}\mu(x)$ on maps $T:X\to Y$ such that $T_{*}\mu=\nu$. A difficulty is that
this problem might be ill posed. For example, if $\mu$ is a Dirac mass and
$\nu$ is not, there is no map $T$ such that $T_{*}\mu=\nu$. Moreover, even
when such maps $T$ exist, the set of such maps does not verify good properties
that allow to apply classical variational methods.

Kantorovitch’s tour de force [126] is twofold. He starts by relaxing the Monge
problem in looking for transport plans. Those are probability measures
$\gamma$ on $X\times Y$ whose marginals are given by $\pi_{1*}\gamma=\mu$ and
$\pi_{2*}\gamma=\nu$. He then wants to minimize $\int_{X\times Y}c(x,y)\ {\rm
d}\gamma(x,y)$ amongst such plans. The set of plans is always nonempty (as
$\mu\otimes\nu$ is one such) and it is closed and convex when $X$ and $Y$ are
compact for instance. Hence existence of an optimal (minimizing) plan can
easily be proved under mild regularity hypotheses on $c$. The second aspect of
his contribution is to provide a dual equivalent problem. The minimal cost of
a transport plan is given by

$\sup\left(\int_{Y}\varphi(y)\ {\rm d}\nu(y)-\int_{X}\psi(x){\rm
d}\mu(x)\right),$

where the supremum is taken amongst pairs of continuous functions
$\varphi:Y\to\mathbb{R}$ and $\psi:X\to\mathbb{R}$ such that

$\forall(x,y)\in X\times Y,\quad\varphi(y)-\psi(x)\leqslant c(x,y).$

It will become clear in the next section that the minimizing problem of
transport plans resembles that of minimizing Mather measures in Aubry–Mather
theory. The dual problem of finding optimal Monge–Kantorovitch pairs of
functions is transparently similar to the notion of subsolutions in weak KAM
theory, especially when both spaces $X$ and $Y$ coincide. Finding a transport
map $T$ amounts to proving that an optimal plan is concentrated on the graph
of a function from $X$ to $Y$. This is analogous to Mather’s Graph Theorem
stating that Mather measures are concentrated on a graph. This deep parallel
was drawn and studied by Bernard and Buffoni [40, 39, 38].

#### Discrete weak KAM theory, with an economical twist

The main idea of discrete weak KAM Theory is to directly discretize the
Lax–Oleinik semigroup, allowing the time to take integer values only (or
integer multiples of a given fixed value). This simple idea then allows to
weaken hypotheses on the phase space. More precisely, coming back to the
definition of the Lax–Oleinik semigroup (1) for $t=1$, setting

$h_{1}(x,y)=\inf_{\begin{subarray}{c}\gamma:[-t,0]\to M\\\ \gamma(0)=x\\\
\gamma(1)=y\end{subarray}}\int_{0}^{1}L\big{(}\gamma(s),\dot{\gamma}(s)\big{)}{\rm
d}s,$

the operator can be rewritten $U(1,x)=\inf\limits_{y\in
X}u_{0}(y)+h_{1}(y,x)$. To write such a formula, very little structure is
needed. Only an underlying space $X$ that we will assume to be metric and a
function $c:X\times X\to\mathbb{R}$ that will play the role of $h_{1}$. A
first theoretical study of such an operator was done in [40]. Further
analogies and results coming from classical weak KAM theory were the subject
of the author’s PhD thesis in which he also highlighted some fundamental
differences.

Here is an economical interpretation of discrete weak KAM theory. Assume that
$X$ is the (uncountable) metric space whose points are wine stores in France.
Let $DRC:X\to\mathbb{R}$ be the function that gives the price $DRC(x)$ of a
bottle of Domaine de la Romanée Conti in the store $x$222Domaine de la Romanée
Conti is maybe the best (as far as comparisons can be made between _œuvres
d’art_) and certainly most prestigious wine in the world and it is a dream of
the author to taste wines from this estate in his life. Unfortunately, in this
randomly chosen example, the function $DRC$ is close to being $+\infty$
everywhere but at Vosne Romanée and accessible at $x=$ Vosne Romanée (but with
no available stock).. Let now $c:X\times X\to\mathbb{R}$ be the function where
$c(x,y)$ is the price for a 24 hour delivery of a wine bottle from $x$ to $y$.
Then if Maxime lives at $x$, the least price he will have to pay to obtain a
bottle of Romanée Conti tomorrow is

$T^{-}DRC(x)=\inf_{y\in X}DRC(y)+c(y,x).$

In this simple and simplistic model, by iterating the previous operator
$T^{-}$, one obtains the best price to have a bottle if one is willing to wait
a long time. Finally, studying long optimal trajectories that a bottle will
follow before reaching the patient Maxime will provide important objects of
weak KAM theory and Aubry–Mather theory.

### Organisation of the text

The first 3 Chapters are dedicated to presenting discrete weak KAM theory in a
general setting. Each of them ends with a section where related classical weak
KAM results are stated to give the reader an overview of the classical theory
without proofs. We believe that this back and forth between the discrete and
the classical weak KAM theories is original. For well chosen costs, it
highlights in a new way strong similarities and also key differences between
the two versions of weak KAM theory. Discrete weak KAM theory as presented in
these Chapters was developed by the author in [174, 173] (following earlier
works as [40]) in a non compact setting. Here we present the compact setting
that is less technical, therefore easier for a first encounter with weak KAM
theory. Yet all key features and ideas of the classical theory persist and are
better highlighted. Further and more precise results written hereafter were
obtained with coauthors, references being provided in the text. Those first
Chapters should be accessible without any specialized background.

The First Chapter introduces the Lax–Oleinik semigroups (negative and
positive). The discrete weak KAM Theorem is proved and the last part of the
Chapter is dedicated to constructing continuous strict subsolutions, which are
a fundamental tool in the theory. This allows to define the Aubry set. In the
last section, a proof of the weak KAM Theorem using the discrete one is given.
Also, it is shown that for a natural cost function, weak KAM solutions and
discrete weak KAM solutions coincide. The result is new to our knowledge. We
also establish that for this cost, the projected Aubry set is equal to the
classical one.

The Second Chapter aims at showing results of a more dynamical nature about
the Aubry set. We introduce Peierls’ barrier and characterize points of the
projected Aubry set with it. Examples of points and chains of the Aubry set
are given. Finally the problem of regularity of subsolutions (or lack thereof)
is addressed. In a first part, in the general setting, we characterize the
projected Aubry set as the set where all subsolutions are continuous. Then by
adding structure, we show existence of $C^{1,1}$ subsolutions, thus obtaining
results similar to Bernard’s classical ones. Finally we provide Graph Theorems
and by adding a twist condition (which replaces convexity) we show how to
define a partial dynamics on the Aubry set.

The Third Chapter is dedicated to Mather measures and to the crucial role they
play in proving convergence of the discounted solutions. We start by giving
two definitions of Mather measures and showing that they are equivalent. Then
we prove convergence of solutions to the discounted equations. It is pointed
out that the limit weak KAM solution for the positive Lax–Oleinik semigroup is
not necessarily the conjugate of the limit weak KAM solution for the negative
Lax–Oleinik semigroup. We then study a degenerate discounted problem that is
new in this setting and prove convergence of the solutions.

The Fourth Chapter provides examples in dimension 1. Those examples come from
autonomous Hamiltonians and have the great merit that explicit computations
can be made. We also show that the weak KAM solutions selected by the
discounted approximation procedure may differ in the discrete setting and in
the classical one. Such examples are folklore to experts. However we do not
know of any reference where a detailed analysis is made under the scope of
weak KAM theory. We believe that they are useful to have in mind in order to
develop an intuition and test conejctures.

The Fifth and last Chapter puts back in the context of discrete weak KAM
theory the foundational problem of conservative twist maps of the annulus. We
revisit results of Mather, Aubry, Bangert… from the perspective of weak KAM
solutions. In this unified setting we gather proofs of well known results that
are spread in various references. We hope this will make them more accessible.
We also give a precise description of what those weak KAM solutions look like
in this setting (results of Arnaud and the author). Finally we conclude with
statements of results or Arnaud and the author opening to the world of weakly
integrable twist maps.

##  Chapter 1 The discrete setting, weak KAM solutions and subsolutions

The main idea of the discrete setting we focus on is to directly discretize
the Lax–Oleinik semigroup, allowing the time to take integer values only (or
integer multiples of a given fixed value). This simple idea then allows to
weaken the assumptions on the phase space. Most results and proofs of this
Chapter are extracted as a particular compact case of [174].

###  1.1 Discrete setting and the Lax–Oleinik semigroup

We focus our attention on the case where $(X,d)$ is a compact metric space.
The analogue of the Lagrangian function is a cost function $c:X\times
X\to\mathbb{R}$ which is assumed to be continuous. One can then define the
Lax–Oleinik semigroup acting on the set of bounded functions
${\mathcal{B}}(X,\mathbb{R})$:

######  Definition 1.1.1.

The Lax–Oleinik semigroup
$T^{-}:{\mathcal{B}}(X,\mathbb{R})\to{\mathcal{B}}(X,\mathbb{R})$ associates
to $f:X\to\mathbb{R}$ the function

$T^{-}f:x\in X\mapsto T^{-}f(x)=\inf_{y\in X}f(y)+c(y,x).$

######  Remark 1.1.2.

The Lax–Oleinik semigroup is a convolution with kernel $c$ in the $(\min,+)$
semiring. The $\inf$ plays the role of integration and the $+$ plays the role
of multiplication in a classical convolution.

In particular, if the set $X$ is finite, the Lax–Oleinik semigroup reduces to
a product (in the $(\min,+)$ semiring) with the matrix whose entries are given
by $c$.

If $f$ is a continuous function, then the infimum in the definition of
$T^{-}f(x)$ is a minimum by compactness of $X$.

We define the sup–norm $\|\cdot\|_{\infty}$ on the space
${\mathcal{B}}(X,\mathbb{R})$ by setting $\|f\|_{\infty}=\sup\limits_{x\in
X}|f(x)|$ for $f\in{\mathcal{B}}(X,\mathbb{R})$. The normed vector space
$\big{(}{\mathcal{B}}(X,\mathbb{R}),\|\cdot\|_{\infty}\big{)}$ is a Banach
space.

We start by listing first basic properties of $T^{-}$:

######  Proposition 1.1.3.

  1. (i)

The image of $T^{-}$ consists of equicontinuous functions with uniformly
bounded amplitude. 111By amplitude of a function $f:X\to\mathbb{R}$ we mean
$\sup f-\inf f$.

  2. (ii)

The Lax–Oleinik semigroup commutes with addition of constant functions: if
$k\in\mathbb{R}$ and $f$ is a function, $T^{-}(f+k)=(T^{-}f)+k$.

  3. (iii)

The Lax–Oleinik semigroup is order preserving: if $f\leqslant g$ then
$T^{-}f\leqslant T^{-}g$.

  4. (iv)

The Lax–Oleinik semigroup is $1$–Lipschitz for the sup–norm
$\|\cdot\|_{\infty}$.

###### Proof.

Let us consider a modulus of uniform continuity $\omega$ for $c$ ($X$ being
compact). This is a non–decreasing function $\omega:[0,+\infty)\to[0,+\infty)$
that is continuous at $0$, with $\omega(0)=0$, such that

$\forall(x,y,x^{\prime},y^{\prime})\in
X^{4},\quad|c(x,y)-c(x^{\prime},y^{\prime})|\leqslant\omega\big{(}d(x,x^{\prime})+d(y,y^{\prime})\big{)}.$

Without loss of generality, by triangular inequality we may assume that
$\omega$ is bounded and that $\|\omega\|_{\infty}\leqslant 2\|c\|_{\infty}$.
Let $f:X\to\mathbb{R}$ be a bounded function and $\varepsilon>0$. Let
$(x,x^{\prime})\in X^{2}$. By definition of the Lax–Oleinik semigroup, there
exists a $y_{\varepsilon}$ such that $T^{-}f(x^{\prime})\geqslant
f(y_{\varepsilon})+c(y_{\varepsilon},x^{\prime})-\varepsilon$. It follows that

$T^{-}f(x)-T^{-}f(x^{\prime})\leqslant
f(y_{\varepsilon})+c(y_{\varepsilon},x)-f(y_{\varepsilon})-c(y_{\varepsilon},x^{\prime})+\varepsilon\leqslant\omega\big{(}d(x,x^{\prime})\big{)}+\varepsilon.$

Letting $\varepsilon\to 0$ yields that
$T^{-}f(x)-T^{-}f(x^{\prime})\leqslant\omega\big{(}d(x,x^{\prime})\big{)}$. As
$x$ and $x^{\prime}$ play symmetrical roles we find that
$|T^{-}f(x)-T^{-}f(x^{\prime})|\leqslant\omega\big{(}d(x,x^{\prime})\big{)}$.
This is $(i)$ as $x,x^{\prime}$ are arbitrary.

Points $(ii)$ and $(iii)$ are obvious from the definition of $T^{-}$ and
automatically imply $(iv)$. Indeed, if $f$ and $g$ are bounded functions, as
$f-\|f-g\|_{\infty}\leqslant g\leqslant f+\|f-g\|_{\infty}$ we obtain

$T^{-}f-\|f-g\|_{\infty}\leqslant T^{-}g\leqslant T^{-}f+\|f-g\|_{\infty}$

which means that $\|T^{-}f-T^{-}g\|_{\infty}\leqslant\|f-g\|_{\infty}$. ∎

######  Remark 1.1.4.

The Lax–Oleinik semigroup can actually be defined on arbitrary functions
$f:X\to\mathbb{R}$ (not necessarily bounded) with the only modification that
$T^{-}f$ can take the value $-\infty$. However, it can be easily checked that,
as $c$ is bounded, this may only happen if $f$ is unbounded from below. In
this case, $T^{-}f$ is identically $-\infty$. Otherwise, the conclusions of
the previous proposition (1.1.3) still hold, with the same proofs.

Now that those properties have been established, let us move on to the weak
KAM theorem.

###  1.2 The weak KAM Theorem and critical subsolutions

In this section, we will introduce and construct some of the most important
objects of weak KAM theory. The first ones are of course weak KAM solutions
and are given by the following theorem:

######  Theorem 1.2.1 (weak KAM).

There exists a unique constant ${c[0]}\in\mathbb{R}$ for which the equation
$u=T^{-}u+{c[0]}$ admits solutions $u:X\to\mathbb{R}$.

######  Remark 1.2.2.

Such functions are then called weak KAM solutions. The constant ${c[0]}$ is
called the critical value.

It is immediate from Proposition 1.1.3 and Remark 1.1.4 that weak KAM
solutions are automatically continuous.

We will give two proofs of the existence part of the weak KAM Theorem. The
first one is similar to the original proof of Fathi ([89, 93]). The second one
is reminiscent of the work of Lions, Papanicolaou and Varadhan ([132]) on
homogenization, that actually appeared prior to the work of Fathi.

The uniqueness of the constant ${c[0]}$ will be established in a second step.

###### First Proof.

Let us introduce
$\mathcal{E}={\mathcal{B}}(X,\mathbb{R})/\mathbb{R}\mathbbm{1}$ the quotient
of bounded functions by constant functions. The set $\mathcal{E}$ is clearly a
vector space. If $f$ is a bounded function, we will denote by $\bar{f}$ its
projection in $\mathcal{E}$. There is an induced norm on $\mathcal{E}$: if
$f:X\to\mathbb{R}$ is a continuous function, denoting $\bar{f}\in\mathcal{E}$
its class in the quotient, we set $\|\bar{f}\|_{0}=\min\\{\|f+k\|_{\infty},\ \
\ k\in\mathbb{R}\\}$. As $T^{-}$ commutes with addition of constant functions,
it induces an operator ${\mathcal{T}}$ on $\mathcal{E}$ defined by
${\mathcal{T}}\bar{f}=\overline{T^{-}f}$ which is independent of the the
representative $f$ in the equivalence class $\bar{f}$. This new operator is
also continuous. Indeed, if $f,g$ are two bounded functions, then for some
suitably chosen constant $k\in\mathbb{R}$,

$\|{\mathcal{T}}\bar{f}-{\mathcal{T}}\bar{g}\|_{0}\leqslant\|T^{-}f-T^{-}g+k\|_{\infty}\leqslant\|f-g+k\|_{\infty}=\|\bar{f}-\bar{g}\|_{0}.$

Moreover, it follows from the fact that $T^{-}$ has values in equicontinuous
functions with uniformly bounded amplitude (see proposition 1.1.3) and from
the Arzelà–Ascoli theorem ([83, Theorem 6.4 page 267]) that
${\mathcal{T}}(\mathcal{E})$ is relatively compact. We can therefore apply the
Schauder–Tychonoff Theorem ([83, Theorems 2.2 and 3.2 pages 414-415]) which
asserts that ${\mathcal{T}}$ has a fixed point. This exactly means that there
exists a bounded function $u:X\to\mathbb{R}$ and a constant $C$ such that
$u=T^{-}u+C$. ∎

The use of the Schauder–Tychonoff Theorem at the end of this first proof,
though natural, is not really necessary. Indeed, fixed points of $1$–Lipschitz
maps can be obtained by much simpler arguments, usually by perturbing the map,
making it contracting and then passing to the limit. This is the spirit of the
second proof in which we use an approximation called discounted procedure:

###### Second Proof.

Let $\lambda\in(0,1)$ and let us introduce the discounted operator
$T^{-}_{\lambda}$ which acts on bounded functions as follows:

$\forall f\in{\mathcal{B}}(X,\mathbb{R}),\ \ \forall x\in X,\quad
T^{-}_{\lambda}f(x)=\inf_{y\in X}\lambda f(y)+c(y,x)=T^{-}(\lambda f)(x).$

Of course, the last formulation, together with the $1$–Lipschitz nature of
$T^{-}$ imply that $T^{-}_{\lambda}$ is now $\lambda$–Lipschitz. Hence, by the
Banach fixed point theorem ([83, Theorem 7.2 page 305]), as
${\mathcal{B}}(X,\mathbb{R})$ is a Banach space, $T^{-}_{\lambda}$ admits a
unique fixed point $u_{\lambda}$ which then verifies
$T^{-}_{\lambda}u_{\lambda}=u_{\lambda}$. By Proposition 1.1.3, the
$(u_{\lambda})_{\lambda\in(0,1)}$ are equicontinuous with uniformly bounded
amplitude, as they all belong to the image of $T^{-}$. Moreover

$\forall x\in X,\quad\min_{(y,z)\in X\times
X}c(y,z)\leqslant(1-\lambda)u_{\lambda}(x)\leqslant\max_{z\in X}c(z,z).$

To prove the left inequality, fix $\lambda$ and take $x_{1}$ such that
$u_{\lambda}(x_{1})$ is minimal. One then has for some $y\in X$,

$u_{\lambda}(x_{1})=\lambda u_{\lambda}(y)+c(y,x_{1})\geqslant\lambda
u_{\lambda}(x_{1})+c(y,x_{1}).$

The right inequality follows from the fact that by definition of
$T^{-}_{\lambda}$ we obviously have
$u_{\lambda}(x)=T^{-}_{\lambda}u_{\lambda}(x)\leqslant\lambda
u_{\lambda}(x)+c(x,x)$.

Let us fix a point $x_{0}\in X$ and define
$\hat{u}_{\lambda}=u_{\lambda}-u_{\lambda}(x_{0})$ for all $\lambda\in(0,1)$.
The previous remarks show that we can find a sequence $\lambda_{n}\to 1$ such
that $(1-\lambda_{n})u_{\lambda_{n}}(x_{0})$ converges to a constant $C$ and
$(\hat{u}_{\lambda_{n}})_{n\in\mathbb{N}}$ uniformly converges to a function
$u$, as $n\to+\infty$. Note that in fact, as the functions $u_{\lambda}$ have
bounded amplitude, one has $(1-\lambda_{n})u_{\lambda_{n}}\to C$ uniformly. It
now follows that

$\forall x\in
X,\quad\hat{u}_{\lambda_{n}}(x)=u_{\lambda_{n}}(x)-u_{\lambda_{n}}(x_{0})\\\
=T^{-}(\lambda_{n}u_{\lambda_{n}})(x)-u_{\lambda_{n}}(x_{0})\\\
=T^{-}(\lambda_{n}\hat{u}_{\lambda_{n}})(x)+(\lambda_{n}-1)u_{\lambda_{n}}(x_{0}).$

Letting $n\to+\infty$, by continuity of $T^{-}$, we conclude that
$u=T^{-}u-C$. ∎

It remains to prove the uniqueness of the critical constant ${c[0]}$. This is
a direct consequence of the following proposition:

######  Proposition 1.2.3.

Assume that $u=T^{-}u+{c[0]}$ for some function $u$ and constant ${c[0]}$,
then

$\forall
v\in{\mathcal{B}}(X,\mathbb{R}),\quad\frac{T^{-n}v}{n}\underset{n\to+\infty}{\longrightarrow}-{c[0]},$

where $T^{-n}=T^{-}\circ\cdots\circ T^{-}$ denotes the $n$-th iterate of the
Lax–Oleinik semigroup, and the convergence is uniform.

###### Proof.

This result is a direct consequence of the non expansive character of the
Lax–Oleinik semigroup. Indeed, by induction, one obtains that for all integers
$n>0$, $u=T^{-n}u+n{c[0]}$. Hence, if $v\in{\mathcal{B}}(X,\mathbb{R})$, then

$\|T^{-n}u-T^{-n}v\|_{\infty}=\|u-n{c[0]}-T^{-n}v\|_{\infty}\leqslant\|u-v\|_{\infty}.$

The result follows immediately as
$\|n{c[0]}+T^{-n}v\|_{\infty}\leqslant\|u-v\|_{\infty}+\|u\|_{\infty}$. ∎

######  Remark 1.2.4.

  1. (i)

In the previous Proposition, one derives that for all
$v\in{\mathcal{B}}(X,\mathbb{R})$ the sequence
$(T^{-n}v+n{c[0]})_{n\in\mathbb{N}}$ is uniformly bounded.

  2. (ii)

It follows from the uniqueness of ${c[0]}$ that in the second proof of the
weak KAM Theorem (1.2.1), the whole family $(1-\lambda)u_{\lambda}$ uniformly
converges to ${c[0]}$ as $\lambda\to 1$.

  3. (iii)

If $u:X\to\mathbb{R}$ is a weak KAM solution, it satisfies the following
fundamental inequalities:

$\forall(x,y)\in X\times X,\quad u(y)-u(x)\leqslant c(x,y)+{c[0]}.$

This last Remark motivates the following definition:

######  Definition 1.2.5.

Given $C\in\mathbb{R}$, a function $v:X\to\mathbb{R}$ will be termed
$C$–subsolution if

$\forall(x,y)\in X\times X,\quad v(y)-v(x)\leqslant c(x,y)+C,$

or equivalently, if $v\leqslant T^{-}v+C$.

We will call ${c[0]}$–subsolutions critical subsolutions, or just subsolutions
when no confusion is possible. We will denote by ${\mathcal{S}}_{C}$ the set
of $C$–subsolutions, and by ${\mathcal{S}}={\mathcal{S}}_{c[0]}$ the set of
critical subsolutions.

Here are some first properties of subsolutions:

######  Proposition 1.2.6.

Given $C\in\mathbb{R}$, the following hold:

  1. (i)

Any $C$–subsolution is bounded.

  2. (ii)

The set ${\mathcal{S}}_{C}$ of $C$–subsolutions is closed (with respect to
pointwise convergence), convex and stable by the Lax–Oleinik semigroup:
$T^{-}({\mathcal{S}}_{C})\subset{\mathcal{S}}_{C}$.

###### Proof.

The first point comes from the fact that if $u\in{\mathcal{S}}_{C}$ and
$x_{0}\in X$, for all $y\in X$,

$u(x_{0})-c(y,x_{0})-C\leqslant u(y)\leqslant u(x_{0})+c(x_{0},y)+C,$

hence $\|u\|_{\infty}\leqslant|u(x_{0})|+\|c+C\|_{\infty}$.

The fact that ${\mathcal{S}}_{C}$ is closed and convex is immediate from the
definition. Only stability deserves an explanation. It follows from the fact
that $u\in{\mathcal{S}}_{C}$ if and only if $u\leqslant T^{-}u+C$ as can be
checked by applying the definitions. Consequently, by the properties of the
Lax–Oleinik semigroup (1.1.3), if $u\leqslant T^{-}u+C$ then

$T^{-}u\leqslant T^{-}(T^{-}u+C)=T^{-}(T^{-}u)+C,$

which means $T^{-}u\in{\mathcal{S}}_{C}$. ∎

We conclude this section by one last characterization of the critical constant
${c[0]}$:

######  Proposition 1.2.7.

The following holds: ${c[0]}=\min\\{C\in\mathbb{R},\ \
{\mathcal{S}}_{C}\neq\varnothing\\}$.

###### Proof.

As follows from the weak KAM Theorem, ${\mathcal{S}}_{c[0]}\neq\varnothing$,
so we just have to prove that if for some constant $C$,
${\mathcal{S}}_{C}\neq\varnothing$, then $C\geqslant{c[0]}$. Let then
$u\in{\mathcal{S}}_{C}$ for some $C\in\mathbb{R}$. As in the last proof, we
get by induction that for all positive integer $n$, $u\leqslant T^{-n}u+nC$.
Therefore, dividing by $n$, we infer that $u/n-C\leqslant T^{-n}u/n$. But the
right hand side converges to $-{c[0]}$ by Proposition 1.2.3. Hence passing to
the limit we conclude that $-C\leqslant-{c[0]}$. ∎

We illustrate once more our model with a random liquid example as in the
introductory section page Discrete weak KAM theory, with an economical twist.
Here again, $X$ is the space of wine stores in France and $c:X\times
X\to\mathbb{R}$ provides the cost $c(x,y)$ of bringing a bottle of wine from
$x$ to $y$ in a day. Let $P:X\to\mathbb{R}$ denote the price $P(x)$ of a
bottle of Petrus222Petrus is a renowned and quite inaccessible wine from the
Bordeaux region in France. More precisely it is the leading estate of the
appellation Pomerol. The royalties earned by a lifetimes’work of the author
would probably allow him to buy a quarter of a bottle of Petrus. at a location
$x$. Let $n\in\mathbb{N}$, then if the author is at $x\in X$, the best price
he will pay to have a bottle of Petrus in $n$ days is $T^{-n}P(x)$.
Proposition 1.2.3 then states that whatever the initial price $P$, for
$n\to+\infty$ the amount $T^{-n}P(x)$ grows like $-nc[0]$. In this particular
example, given the order of magnitude of $P$, the time $n$ would have to be
very very large to compensate the initial price.

###  1.3 The positive Lax–Oleinik semigroup

In this section, we introduce the positive Lax–Oleinik semigroup and state its
main properties. As they are analogous to the properties of the (negative)
Lax–Oleinik semigroup, the proofs are omitted and left to the reader.

######  Definition 1.3.1.

The positive Lax–Oleinik semigroup
$T^{+}:{\mathcal{B}}(X,\mathbb{R})\to{\mathcal{B}}(X,\mathbb{R})$ maps to
$f:X\to\mathbb{R}$ the function

$T^{+}f:x\in X\mapsto T^{+}f(x)=\sup_{y\in X}f(y)-c(x,y).$

######  Remark 1.3.2.

From the definition, one checks that
$T^{+}f=-\big{(}T^{-}_{\hat{c}}(-f)\big{)}$ where $T^{-}_{\hat{c}}$ is the
cost defined by $\hat{c}(x,y)=c(y,x)$. Hence the fact that $T^{-}$ and $T^{+}$
share very similar properties is not surprising.

######  Proposition 1.3.3.

  1. (i)

The image of $T^{+}$ consists of equicontinuous functions with uniformly
bounded amplitude.

  2. (ii)

The positive Lax–Oleinik semigroup, $T^{+}$, commutes with addition of
constant functions.

  3. (iii)

The positive Lax–Oleinik semigroup is order preserving:

if $f\leqslant g$ then $T^{+}f\leqslant T^{+}g$.

  4. (iv)

The positive Lax–Oleinik semigroup is $1$–Lipschitz for the sup–norm
$\|\cdot\|_{\infty}$.

The positive semigroup also fulfills a weak KAM Theorem:

######  Theorem 1.3.4 (positive weak KAM).

The critical constant ${c[0]}$ is the unique real value $c\in\mathbb{R}$ for
which the equation $u=T^{+}u-c$ admits solutions $u:X\to\mathbb{R}$. Moreover
the constant $c[0]$ has the following caracterization:

$\forall
v\in{\mathcal{B}}(X,\mathbb{R}),\quad\frac{T^{+n}v}{n}\underset{n\to+\infty}{\longrightarrow}{c[0]},$

where $T^{+n}=T^{+}\circ\cdots\circ T^{+}$ denotes the $n$-th iterate of the
positive Lax–Oleinik semigroup, and the convergence is uniform.

######  Remark 1.3.5.

Such functions are then called positive weak KAM solutions.

It is immediate from Proposition 1.3.3 that positive weak KAM solutions are
always continuous.

In the positive weak KAM theorem, the fact that the critical constant for
$T^{+}$ is the same as that of $T^{-}$, that is ${c[0]}$, deserves some
explanation. It is checked that $u:X\to\mathbb{R}$ is a $C$–subsolution, for
some $C\in\mathbb{R}$ if and only if $u\geqslant T^{+}u-C$. Hence, as for the
Lax–Oleinik semigroup, one has that
$T^{+}({\mathcal{S}}_{C})\subset{\mathcal{S}}_{C}$.

Now, the fact that both critical constants coincide follows from Proposition
1.2.7 which characterizes the critical constant only making use of the notion
of subsolution, and not of either the positive, nor negative Lax–Oleinik
semigroup. However, its proof can be done equivalently using the negative or
the positive Lax–Oleinik semigroup.

We end this section with a curiosity on the composition of positive and
negative Lax–Oleinik semigroups:

######  Proposition 1.3.6.

Let $u\in{\mathcal{B}}(X,\mathbb{R})$, then $T^{+}\circ T^{-}u\leqslant u$ and
$T^{-}\circ T^{+}u\geqslant u$.

###### Proof.

Let us establish the first inequality. If $x\in X$ then

$T^{+}\circ T^{-}u(x)=\sup_{y\in X}T^{-}u(y)-c(x,y)=\sup_{y\in X}\inf_{z\in
X}u(z)+c(z,y)-c(x,y)\leqslant u(x),$

where the last inequality is obtained by taking $z=x$. ∎

###  1.4 Strict subsolutions, Aubry sets

In this section, we will focus our study on critical subsolutions, hence the
adjective may be omitted from time to time, but is always implicit. Recall
that the set ${\mathcal{S}}$ is the set of critical subsolutions. The goal
will be to construct a special kind of subsolutions which are in some sense
better than the others:

######  Theorem 1.4.1.

There exists a subsolution $u_{0}\in{\mathcal{S}}\cap C^{0}(X,\mathbb{R})$
such that, if the equality $u_{0}(y)-u_{0}(x)=c(x,y)+{c[0]}$ holds for some
$(x,y)\in X\times X$, then

$\forall u\in{\mathcal{S}},\quad u(y)-u(x)=c(x,y)+{c[0]}.$ (1.1)

A subsolution verifying this last property (1.1) will be termed strict. The
proof of this Theorem will occupy the rest of this section. It will be divided
in two main parts. In a first one, we will construct the function $u_{0}$ and
prove that it verifies (1.1) for all other continuous subsolutions.

Then a digression is devoted to studying the structure of the set where the
equality $u_{0}(y)-u_{0}(x)=c(x,y)+{c[0]}$ takes place. This is the Aubry set,
a central object in weak KAM theory, and unsurprisingly in Aubry–Mather
theory.

This being achieved, the proof of the Theorem ends rather easily.

###### Beginning of the Proof.

The set $(X,d)$ being metric and compact, the Banach space
$\big{(}C^{0}(X,\mathbb{R}),\|\cdot\|_{\infty}\big{)}$ is itself separable.
Therefore, ${\mathcal{S}}\cap C^{0}(X,\mathbb{R})$ being a subset of a
separable space is also separable. Let $(v_{n})_{n\in\mathbb{N}}$ be a dense
(with respect to the topology of uniform convergence) sequence in
${\mathcal{S}}\cap C^{0}(X,\mathbb{R})$. Now, let us define
$u_{0}=\sum\limits_{n\geqslant 0}a_{n}v_{n}$ where for $n>0$, we set
$a_{n}=\min(2^{-n},2^{-n}/\|v_{n}\|_{\infty})$ and
$a_{0}=1-\sum\limits_{n>0}a_{n}$.

The function $u_{0}$ is defined by a series of continuous functions that
converges for the $\|\cdot\|_{\infty}$–norm, hence it is continuous. It is an
infinite convex combination of critical subsolutions, therefore, by
Proposition 1.1.3 it is a subsolution. We will prove it verifies the property
we seek for.

Let us consider now $(x,y)\in X\times X$ such that
$u_{0}(y)-u_{0}(x)=c(x,y)+{c[0]}$. By definition of subsolutions, we know that
$v_{n}(y)-v_{n}(x)\leqslant c(x,y)+{c[0]}$, for all $n\geqslant 0$.
Multiplying each of these inequalities by $a_{n}$ and summing, we get

$u_{0}(y)-u_{0}(x)=\sum_{n=0}^{+\infty}a_{n}\big{(}v_{n}(y)-v_{n}(x)\big{)}\leqslant\sum_{n=0}^{+\infty}a_{n}(c(x,y)+{c[0]})=c(x,y)+{c[0]},$

which is in fact an equality. Hence all the middle inequalities must be
equalities, and the $a_{n}$ being all positive, we conclude that
$v_{n}(y)-v_{n}(x)=c(x,y)+{c[0]}$ for all integer $n$. The sequence
$(v_{n})_{n\in\mathbb{N}}$ being dense in ${\mathcal{S}}\cap
C^{0}(X,\mathbb{R})$ we conclude eventually that

$\forall u\in{\mathcal{S}}\cap
C^{0}(X,\mathbb{R}),\quad\Big{[}u_{0}(y)-u_{0}(x)=c(x,y)+{c[0]}\Big{]}\Longrightarrow\Big{[}u(y)-u(x)=c(x,y)+{c[0]}\Big{]}.$

∎

The rest of the proof will consist in extending this property to non
continuous subsolutions.

Let us start by defining some useful sets:

######  Definition 1.4.2 (Aubry sets).

Let $u\in{\mathcal{S}}$ be a critical subsolution.

  * •

The non–strict set of $u$ is

$\mathcal{NS}_{u}=\\{(x,y)\in X\times X,\quad u(y)-u(x)=c(x,y)+{c[0]}\\}.$

  * •

The Aubry set of $u$ is

$\widetilde{\mathcal{A}}_{u}=\Big{\\{}(x_{n})_{n\in\mathbb{Z}}\in
X^{\mathbb{Z}},\ \ \forall n<p,\quad
u(x_{p})-u(x_{n})=\sum_{k=n}^{p-1}c(x_{k},x_{k+1})+(n-p){c[0]}\Big{\\}}.$

  * •

The Aubry set is
$\widetilde{\mathcal{A}}=\widetilde{\mathcal{A}}_{u_{0}}\subset
X^{\mathbb{Z}}$ where $u_{0}$ is the peculiar subsolution previously
constructed.

  * •

The $2$–Aubry is $\widehat{\mathcal{A}}=\mathcal{NS}_{u_{0}}\subset X\times
X$.

  * •

Finally, the projected Aubry set is
${\mathcal{A}}=\pi_{1}(\widehat{\mathcal{A}})\subset X$ where $\pi_{1}:X\times
X\to X$ is the projection on the first factor: $(x,y)\mapsto x$.

######  Remark 1.4.3.

  1. (i)

The beginning of the proof of Theorem 1.4.1 may be summed up in the equality
$\widehat{\mathcal{A}}=\bigcap\limits_{u\in{\mathcal{S}}\cap
C^{0}(X,\mathbb{R})}\mathcal{NS}_{u}$.

  2. (ii)

If $s:X^{\mathbb{Z}}\to X^{Z}$ is the shift operator:
$(x_{n})_{n\in\mathbb{Z}}\mapsto(x_{n+1})_{n\in\mathbb{Z}}$ then the sets
introduced above are invariant by this shift:
$\widetilde{\mathcal{A}}_{u}=s(\widetilde{\mathcal{A}}_{u})$ and
$\widetilde{\mathcal{A}}=s(\widetilde{\mathcal{A}})$.

  3. (iii)

If $u\in{\mathcal{S}}$ and $(x_{n})_{n\in\mathbb{Z}}\in X^{\mathbb{Z}}$ then
one has for all $k\in\mathbb{Z}$, $u(x_{k+1})-u(x_{k})\leqslant
c(x_{k},x_{k+1})+{c[0]}$. Summing these inequalities, one gets

$\forall n<p,\quad
u(x_{p})-u(x_{n})\leqslant\sum_{k=n}^{p-1}c(x_{k},x_{k+1})+(n-p){c[0]}.$

Therefore if the previous inequality turns out to be an equality, it implies
that $u(x_{k+1})-u(x_{k})=c(x_{k},x_{k+1})+{c[0]}$ for $n\leqslant k<p$.

  4. (iv)

Note that by taking $n=0$ and $p=1$ in the definition of
$\widetilde{\mathcal{A}}_{u}$ we find that
$\pi_{0,1}(\widetilde{\mathcal{A}}_{u})\subset\mathcal{NS}_{u}$ where
$\pi_{0,1}:X^{\mathbb{Z}}\to X\times X$ is the projection:
$(x_{n})_{n\in\mathbb{Z}}\mapsto(x_{0},x_{1})$.

Now let us study these Aubry sets. We start by basic topological properties:

######  Proposition 1.4.4.

The $2$–Aubry set $\widehat{\mathcal{A}}$ is closed and non–empty.

###### Proof.

Being closed comes from the identity $\widehat{\mathcal{A}}=F^{-1}\\{0\\}$
where $F(x,y)=u_{0}(y)-u_{0}(x)-c(x,y)-{c[0]}$ is continuous.

Being non–empty is a consequence of the minimality of ${c[0]}$ (Proposition
1.2.7). Indeed, as $u_{0}$ is a subsolution, the function $F$ is non–positive.
By compactness and continuity, if $\widehat{\mathcal{A}}=F^{-1}\\{0\\}$ were
empty, there would be a small $\varepsilon>0$ such that
$u_{0}(y)-u_{0}(x)\leqslant c(x,y)+{c[0]}-\varepsilon$, for all $(x,y)\in
X\times X$. But this means that
$u_{0}\in{\mathcal{S}}_{{c[0]}-\varepsilon}\neq\varnothing$ which contradicts
Proposition 1.2.7. ∎

The next proposition states that elements of the $2$–Aubry set come in
families, meaning that the Aubry set is not empty:

######  Proposition 1.4.5.

Let $(x,y)\in\widehat{\mathcal{A}}$, then there exists a sequence
$(x_{n})_{n\in\mathbb{Z}}\in\widetilde{\mathcal{A}}$ such that
$(x,y)=\pi_{0,1}\big{(}(x_{n})_{n\in\mathbb{Z}}\big{)}$. In particular, the
Aubry set is itself closed (for the product topology), not empty, and
$\widehat{\mathcal{A}}=\pi_{0,1}(\widetilde{\mathcal{A}})$.

The proof will make use of the following two lemmas. The first one’s proof is
a direct application of the definitions and is omitted:

######  Lemma 1.4.6.

Let $n$ be a positive integer and $f\in{\mathcal{B}}(X,\mathbb{R})$, then

$\forall x\in X,\quad
T^{-n}f(x)=\inf_{x_{-n},\cdots,x_{0}=x}f(x_{-n})+\sum_{k=-n}^{-1}c(x_{k},x_{k+1}),$
$\forall x\in X,\quad
T^{+n}f(x)=\sup_{x=x_{0},\cdots,x_{n}}f(x_{n})-\sum_{k=0}^{n-1}c(x_{k},x_{k+1}).$

######  Lemma 1.4.7.

Let $u\in{\mathcal{S}}$ be a continuous subsolution and
$(x,y)\in\widehat{\mathcal{A}}$. We have $T^{-n}u(x)=u(x)-n{c[0]}$ and
$T^{+n}u(y)=u(y)+n{c[0]}$ for all positive integer $n$.

###### Proof.

As $T^{-n}u$ and $T^{+n}u$ are continuous subsolutions and
$(x,y)\in\widehat{\mathcal{A}}$, we now know that
$T^{-n}u(y)-T^{-n}u(x)=T^{+n}u(y)-T^{+n}u(x)=c(x,y)+{c[0]}$. It implies
readily that

$T^{-(n+1)}u(y)+{c[0]}\geqslant T^{-n}u(y)=T^{-n}u(x)+c(x,y)+{c[0]}\geqslant
T^{-(n+1)}u(y)+{c[0]},$

the first inequality coming from the fact that $T^{-n}u\in{\mathcal{S}}$ the
second from the definition of $T^{-}$. Hence all inequalities turn out to
equalities and the sequence $(T^{-n}u(y)+n{c[0]})_{n\in\mathbb{N}}$ is
constant.

The same holds for $T^{-n}u(x)+n{c[0]}=T^{-n}u(y)+n{c[0]}-c(x,y)-{c[0]}$.

The proof of the rest of the lemma is established similarly.

∎

###### Proof of Proposition 1.4.5.

The sequences $(x_{n})$ for $n\leqslant 0$ and $n\geqslant 0$ are constructed
separately. The first half will come from the negative Lax–Oleinik semigroup,
the second one, from the positive Lax–Oleinik semigroup.

As $u_{0}$ and $c$ are continuous and $X$ is compact, any supremum (resp.
infimum) involving them is actually a maximum (resp. minimum). Therefore, for
each positive $n$, there exist chains
$x_{-n}^{n},\cdots,x_{0}^{n}=x,x_{1}^{n}=y,\cdots x_{n+1}^{n}$ such that

$T^{-n}u_{0}(x)=u_{0}(x_{-n}^{n})+\sum_{k=-n}^{-1}c(x_{k}^{n},x_{k+1}^{n})\ \
;\quad
T^{+n}u_{0}(y)=u_{0}(x^{n}_{n+1})-\sum_{k=1}^{n}c(x_{k}^{n},x_{k+1}^{n}).$

By a diagonal argument, let $(N_{n})_{n\geqslant 0}$ be an extracted sequence
such that for all $k\geqslant 0$ the sequences $(x_{k}^{N_{n}})_{n\geqslant
k}$ and $(x_{-k}^{N_{n}})_{n\geqslant k}$ converge. We will denote by $x_{k}$
and $x_{-k}$ the respective limits. Obviously, $x_{0}=x$ and $x_{1}=y$. By
definition of the Lax–Oleinik semigroup and Lemma 1.4.7, for all $k\leqslant
n$,

$u_{0}(x)=u_{0}(x_{k}^{N_{n}})+\sum_{i=-k}^{-1}c(x_{i}^{N_{n}},x_{i+1}^{N_{n}})+k{c[0]}\
;$
$u_{0}(y)=u_{0}(x^{N_{n}}_{k+1})-\sum_{i=1}^{k}c(x_{i}^{N_{n}},x_{i+1}^{N_{n}})-k{c[0]}.$

Letting $n\to+\infty$, we obtain that

$u_{0}(x)=u_{0}(x_{k})+\sum_{i=-k}^{-1}c(x_{i},x_{i+1})+k{c[0]}\ \ ;\quad
u_{0}(y)=u_{0}(x_{k+1})-\sum_{i=1}^{k}c(x_{i},x_{i+1})-k{c[0]}.$

As in Remark 1.4.3, this implies that for all $n\in\mathbb{Z}$,
$u_{0}(x_{n+1})-u_{0}(x_{n})=c(x_{n},x_{n+1})+{c[0]}$ (the case $n<0$ is given
by the left equalities and $n>0$ by the right equalities above, $n=0$ is
because $(x,y)\in\widehat{\mathcal{A}}$). Now, again as in Remark 1.4.3, by
summing those equalities, one obtains that

$\forall n<p,\quad
u_{0}(x_{p})-u_{0}(x_{n})=\sum_{k=n}^{p-1}c(x_{k},x_{k+1})+(n-p){c[0]}.$

This exactly means that $(x_{n})_{n\in\mathbb{Z}}\in\widetilde{\mathcal{A}}$.
∎

######  Remark 1.4.8.

Let $(x_{n})_{n\in\mathbb{Z}}\in\widetilde{\mathcal{A}}$ and $u$ be a
subsolution. As for all $n\in\mathbb{Z}$,
$(x_{n},x_{n+1})\in\widehat{\mathcal{A}}$, summing equalities
$u(x_{n+1})-u(x_{n})=c(x_{n},x_{n+1})+{c[0]}$, one finds that

$\forall n<p,\quad
u(x_{p})-u(x_{n})=\sum_{k=n}^{p-1}c(x_{k},x_{k+1})+(n-p){c[0]}.$

In other terms, $\widetilde{\mathcal{A}}\subset\widetilde{\mathcal{A}}_{u}$.

As the Aubry set is invariant by the shift, we get the immediate:

######  Corollary 1.4.9.

The projected Aubry set can be obtained by either projection,
${\mathcal{A}}=\pi_{1}(\widehat{\mathcal{A}})=\pi_{2}(\widehat{\mathcal{A}})$,
where $\pi_{2}$ is the projection on the second factor.

Let us now complete the proof of the Theorem:

###### End of the Proof of Theorem 1.4.1.

Let $u\in{\mathcal{S}}$ be any subsolution. Let
$(x,y)\in\widehat{\mathcal{A}}$ and let
$(x_{n})_{n\in\mathbb{Z}}\in\widetilde{\mathcal{A}}$ such that
$\pi_{0,1}\big{(}(x_{n})_{n\in\mathbb{Z}}\big{)}=(x,y)$. We have the following
chain of inequalities, for all integer $n\in\mathbb{Z}$:

$T^{-}u(x_{n+1})\leqslant u(x_{n})+c(x_{n},x_{n+1})\leqslant
T^{-}u(x_{n})+c(x_{n},x_{n+1})+{c[0]}=T^{-}u(x_{n+1}),$
$T^{+}u(x_{n-1})\geqslant u(x_{n})-c(x_{n-1},x_{n})\geqslant
T^{+}u(x_{n})-c(x_{n-1},x_{n})-{c[0]}=T^{+}u(x_{n-1}).$

In each line, the first inequality is by definition of the Lax–Oleinik
semigroups, the second holds because $u\in{\mathcal{S}}$ and the last equality
comes from the fact that, both $T^{-}u$ and $T^{+}u$ being continuous
subsolutions, the first part of the proof of Theorem 1.4.1 applies.

Hence all inequalities are equalities, and taking $n=0,1$ it follows both
equalities $u(x)=T^{-}u(x)+{c[0]}$ and $u(y)=T^{-}u(y)+{c[0]}$ and eventually
that

$u(y)-u(x)=T^{-}u(y)-T^{-}u(x)=c(x,y)+{c[0]}.$

This ends the proof. ∎

Let us conclude by a corollary of this proof:

######  Corollary 1.4.10.

Let $x\in X$. The following assertions are equivalent:

  1. (i)

$x\in{\mathcal{A}}$,

  2. (ii)

$\forall u\in{\mathcal{S}},\ \ u(x)=T^{-}u(x)+{c[0]}$,

  3. (iii)

$\forall u\in{\mathcal{S}},\ \ u(x)=T^{+}u(x)-{c[0]}$,

  4. (iv)

for any $u\in{\mathcal{S}}$ and $n>0$,
$u(x)=T^{-n}u(x)+n{c[0]}=T^{+n}u(x)-n{c[0]}$.

###  1.5 Relations to the classical theory

####  1.5.1 Classical setting and Lax-Oleinik semigroup

Classical weak KAM theory takes place originally in a smooth, connected and
compact manifold $M$. We will denote by $TM$ the tangent bundle of $M$ and
denote points in this set by $(x,v)\in TM$, where $x\in M$ and $v\in T_{x}M$
is a vector tangent to $M$ at $x$. Similarly, $T^{*}M$ is the cotangent bundle
of $M$, and a point of this cotangent bundle will be written $(x,p)\in
T^{*}M$, where $x\in M$ and $p\in T^{*}_{x}M$ is a linear form on $T_{x}M$.
For convenience, we will equip $TM$ with a Riemannian metric and denote by
$(x,v)\mapsto\|v\|_{x}$ the associated norm. As $M$ is compact, all Riemannian
metrics are equivalent and all results are independent of this choice. The
induced distance on $M$ will be denoted by ${\rm d}(\cdot,\cdot)$.333A simple
example to keep in mind is that of the flat torus
$M=\mathbb{T}^{n}=\mathbb{R}^{n}/\mathbb{Z}^{n}$. In this case, both $TM$ and
$T^{*}M$ are isomorphic to $\mathbb{T}^{n}\times\mathbb{R}^{n}$. As a
Riemannian metric, one may use the canonical Euclidian scalar product on
$\mathbb{R}^{n}$ to define a metric both on $T\mathbb{T}^{n}$ and on
$T^{*}\mathbb{T}^{n}$.  One considers a Tonelli Hamiltonian, that is a
function $H:T^{*}M\to\mathbb{R}$ defined on the cotangent bundle of $M$
verifying the following set of conditions:

  * •

$H$ is $C^{2}$,

  * •

$H$ is strictly convex in the momentum variable, meaning that for any
$(x,p)\in T^{*}M$ the Hessian $\partial_{pp}H(x,p)$ is positive definite.

  * •

$H$ is superlinear, meaning that

$\forall x\in
M,\quad\lim_{\|p\|_{x}\to+\infty}\frac{H(x,p)}{\|p\|_{x}}=+\infty.$

Note that in the superlinearity condition the limit is automatically uniform
in $x$, thanks to the convexity of $H$ and to the compactness of $M$.
Moreover, this condition depends at first sight on the choice of the
Riemannian metric on $TM$, the norm of $p\in T^{*}_{x}M$ being the operator
norm of $p$ and denoted again $\|p\|_{x}$ to simplify notations. However, $M$
being compact, any two Riemannian metrics are equivalent, hence the notion of
superlinearity becomes independent of the initial choice.

Given this Hamiltonian, one studies two equations called Hamilton–Jacobi
equations. The evolutionary Hamilton–Jacobi equation is:

$\begin{cases}\partial_{t}u+H(x,\partial_{x}u)=0,\\\
u(0,\cdot)=u_{0}.\end{cases}$ (EHJ)

Above, the unknown function is $u(t,x):[0,+\infty)\times M\to\mathbb{R}$ and
$u_{0}:M\to\mathbb{R}$ is a given continuous function called initial
condition.

The stationary Hamilton–Jacobi equation is

$H(x,D_{x}u)=\alpha,$ (SHJ)

where the unknown is $u:M\to\mathbb{R}$ and $\alpha\in\mathbb{R}$ is a given
constant.

Strong ($C^{1}$) solutions to those equations rarely exist. For instance, if
$u_{0}$ is smooth, one can solve the evolutionary equation by using the method
of characteristics. However, shocks appear almost systematically in finite
time and the solution ceases to be smooth (results about this can be found in
[93, 81]). Therefore a notion of weak solutions is required and the one we
retain is that of viscosity solution. We provide it for the evolutionary
equation even though it is not explicitly needed. It is left to the reader to
infer the analogous definition for the stationary equation. A good
introduction to the subject is [24]:

######  Definition 1.5.1.

  * •

A continuous function $u:[0,+\infty)\times M\to\mathbb{R}$ is a viscosity
subsolution to (EHJ) if it verifies the initial condition and if for any
$C^{1}$ function $\phi:(0,+\infty)\times M\to\mathbb{R}$, if $u-\phi$ has a
local maximum at $(t_{0},x_{0})$ then

$\partial_{t}\phi(t_{0},x_{0})+H\big{(}x_{0},\partial_{x}\phi(t_{0},x_{0})\big{)}\leqslant
0.$

  * •

A continuous function $u:[0,+\infty)\times M\to\mathbb{R}$ is a viscosity
supersolution to (EHJ) if it verifies the initial condition and if for any
$C^{1}$ function $\phi:(0,+\infty)\times M\to\mathbb{R}$, if $u-\phi$ has a
local minimum at $(t_{0},x_{0})$ then

$\partial_{t}\phi(t_{0},x_{0})+H\big{(}x_{0},\partial_{x}\phi(t_{0},x_{0})\big{)}\geqslant
0.$

  * •

A continuous function $u:[0,+\infty)\times M\to\mathbb{R}$ is a viscosity
solution to (EHJ) if it is both a subsolution and a supersolution.

In the rest of the exposition, unless otherwise specified, any solution,
subsolution or supersolution will be implicitly understood in the viscosity
sense and the adjective will be omitted.

This notion is particularly adapted to our problem for several reasons. The
first one is that it is reasonable in the sense that if a solution $u$ is
differentiable at $(t_{0},x_{0})\in(0,+\infty)\times M$ then it solves the
Hamilton–Jacobi equation at that point:

$\partial_{t}u(t_{0},x_{0})+H\big{(}x_{0},\partial_{x}u(t_{0},x_{0})\big{)}=0.$

As it can be proved that in our setting, solutions are locally Lipschitz,
viscosity solutions turn out to be almost everywhere solutions. However,
beware that the converse is not true. The following theorem makes viscosity
solutions particularly handy (see [24, 81]):

######  Theorem 1.5.2.

Given a continuous function $u_{0}:M\to\mathbb{R}$, there exists a unique
solution to (EHJ). This solution will be denoted $(x,t)\mapsto
S^{-}(t)(u_{0})(x)$.

For any fixed $t>0$, the operator $S^{-}(t)$ is acting on
$C^{0}(M,\mathbb{R})$. Due to the uniqueness of solutions and to the fact that
$H$ is autonomous, $S^{-}$ is a semigroup, meaning that
$S^{-}(t+s)=S^{-}(t)\circ S^{-}(s)$. It turns out it enjoys properties very
similar to the discrete Lax–Oleinik semigroup $T^{-}$:

######  Proposition 1.5.3.

  1. 1.

For any $t>0$, there exists $K>0$ such that the set
$S^{-}(t)\big{(}C^{0}(M,\mathbb{R})\big{)}$ contains only $K$–Lipschitz
functions.

  2. 2.

For any $t>0$, $S^{-}(t)$ commutes with addition of constants:
$S^{-}(t)(f+k)=S^{-}(t)(f)+k$, for all $f\in C^{0}(M,\mathbb{R})$ and
$k\in\mathbb{R}$.

  3. 3.

For any $t>0$, $S^{-}(t)$ is order preserving: if $f\leqslant g$ then
$S^{-}(t)f\leqslant S^{-}(t)g$.

  4. 4.

For any $t>0$, $S^{-}(t)$ is $1$–Lipschitz for the sup–norm.

At this stage, the similarities between discrete weak KAM theory and the
Hamilton–Jacobi equations are not clear. It comes from an explicit
control–theoretic representation formula of the operators $S^{-}(t)$. Let us
define the Lagrangian as the convex dual of the Hamiltonian:

######  Definition 1.5.4.

The Lagrangian $L:TM\to\mathbb{R}$ is defined by

$\forall(x,v)\in TM,\quad L(x,v)=\sup_{p\in T^{*}_{x}M}p(v)-H(x,p).$

In this definition, the supremum is actually a maximum. The Lagrangian $L$ is
termed Tonelli Lagrangian as it enjoys the same properties as $H$:

  * •

$L$ is $C^{2}$,

  * •

$L$ is strictly convex in the speed variable, meaning that for any $(x,v)\in
TM$ the Hessian $\partial_{vv}L(x,v)$ is positive definite.

  * •

$L$ is superlinear, meaning that

$\forall x\in
M,\quad\lim_{\|v\|_{x}\to+\infty}\frac{L(x,v)}{\|v\|_{x}}=+\infty.$

Through the Lagrangian and Hamiltonian functions we can go from $TM$ to
$T^{*}M$ thanks to the Fenchel transform ${\mathcal{L}}$ defined by

$\forall(x,v)\in
TM,\quad{\mathcal{L}}(x,v)=\big{(}x,\partial_{v}L(x,v)\big{)}\in T^{*}M.$
(1.2)

This transformation is a $C^{1}$ diffeomorphism under the Tonelli assumptions
and its inverse is given by

$\forall(x,p)\in
T^{*}M,\quad{\mathcal{L}}^{-1}(x,p)=\big{(}x,\partial_{p}H(x,p)\big{)}\in TM.$

Moreover, $H$ and $L$ are also related by the formulas

$H\big{(}x,\partial_{v}L(x,v)\big{)}=\partial_{v}L(x,v)(v)-L(x,v)\ \ ;\ \
L\big{(}x,\partial_{p}H(x,p)\big{)}=p\big{(}\partial_{p}H(x,p)\big{)}-H(x,p).$

######  Theorem 1.5.5.

Let $u:M\to\mathbb{R}$ be any continuous function. For any $t>0$ and $x\in M$
the following holds:

$S^{-}(t)u(x)=\inf_{\begin{subarray}{c}\gamma:[-t,0]\to M\\\
\gamma(0)=x\end{subarray}}u\big{(}\gamma(-t)\big{)}+\int_{-t}^{0}L\big{(}\gamma(s),\dot{\gamma}(s)\big{)}{\rm
d}s.$ (1.3)

In this formula, called the Lax–Oleinik formula, the infimum is taken amongst
absolutely continuous curves. Tonelli theory asserts that the infimum is a
minimum and any such minimum turns out to be $C^{2}$ and to verify the
Euler–Lagrange equation: $\dfrac{{\rm d}}{{\rm
d}t}\partial_{v}L(\gamma,\dot{\gamma})=\partial_{x}L(\gamma,\dot{\gamma})$
(see [68]). This equation defines a complete flow on $TM$ denoted by
$\varphi_{L}$. It is called the Euler–Lagrange flow. Its conjugate by the
Fenchel transform
$\varphi_{H}={\mathcal{L}}\circ\varphi_{L}\circ{\mathcal{L}}^{-1}$ is a flow
on $T^{*}M$ called Hamiltonian flow. Its trajectories solve Hamilton’s
equations:

$\begin{cases}\dot{x}=\partial_{p}H(x,p),\\\
\dot{p}=-\partial_{x}H(x,p).\end{cases}$ (1.4)

The infimum in (1.3) can be split in two by first choosing a starting point
$y$ for the curves and then minimizing between $y$ and $x$. More precisely, if
we define the action functional

$\forall(t,y,x)\in[0,+\infty)\times M\times M,\quad
h_{t}(y,x)=\inf_{\begin{subarray}{c}\gamma:[-t,0]\to M\\\ \gamma(0)=x\\\
\gamma(-t)=y\end{subarray}}\int_{-t}^{0}L\big{(}\gamma(s),\dot{\gamma}(s)\big{)}{\rm
d}s,$ (1.5)

then the formula for solutions of (EHJ) becomes

$S^{-}(t)u(x)=\inf_{y\in M}u(y)+h_{t}(y,x),$

which is exactly the discrete Lax–Oleinik semigroup with cost funtion
$c=h_{t}$.

####  1.5.2 The weak KAM Theorem and critical subsolutions

Another important fact follows from the simple remark that a function
$u:M\to\mathbb{R}$ is solution to the stationary Hamilton–Jacobi equation
(SHJ) with constant $\alpha$ if and only if the function $U(t,x)=u(x)-\alpha
t$ is solution to the evolutionary equation (EHJ) with initial condition
$u_{0}=u$. Hence any such solution is characterized by the property

$\displaystyle\forall(t,x)\in(0,+\infty)\times M,\quad u(x)$
$\displaystyle=\inf_{\begin{subarray}{c}\gamma:[-t,0]\to M\\\
\gamma(0)=x\end{subarray}}u\big{(}\gamma(-t)\big{)}+\int_{-t}^{0}\Big{[}L\big{(}\gamma(s),\dot{\gamma}(s)\big{)}+\alpha\Big{]}{\rm
d}s$ $\displaystyle=\inf_{y\in M}u(y)+h_{t}(y,x)+t\alpha.$

Note also that such solutions verify in particular that

$\forall t>0,\ \forall\gamma:[-t,0]\to M,\quad
u\big{(}\gamma(0)\big{)}-u\big{(}\gamma(-t)\big{)}\leqslant\int_{-t}^{0}\Big{[}L\big{(}\gamma(s),\dot{\gamma}(s)\big{)}+\alpha\Big{]}{\rm
d}s.$ (1.6)

In fact, verifying the above family of inequalities characterizes $u$ to be a
subsolution of (SHJ).

With these facts in mind, it should not come as a surprise that the original
weak KAM Theorem of Fathi ([89]) is similar to the discrete weak KAM Theorem
we stated:

######  Theorem 1.5.6.

There exists a unique constant $\alpha(0)\in\mathbb{R}$ for which the
stationary equation (EHJ) admits a solution with right hand side equal to
$\alpha=\alpha(0)$.444The notation $\alpha(0)$ is borrowed from Mather’s
$\alpha$ function. It is a function acting on the first cohomology group of
$M$. Given a closed $1$–form $c$, one can perturb the stationary equation
(SHJ) by $H(x,c(x)+D_{x}u)=\alpha$ and prove a weak KAM theorem for this
equation. The critical constant found depends only on the cohomology class
$[c]$ and is $\alpha([c])$. Discrete analogues of this are discussed in [173].

###### Proof.

For fun’s sake, we provide yet another proof of this Theorem. We deduce it
from the discrete weak KAM theorem, although it is highly unnatural.

Uniqueness of $\alpha(0)$ follows from the uniqueness of the critical constant
in the discrete weak KAM theorem 1.2.1 as a classical weak KAM solution is
also a discrete weak KAM solution for the cost $h_{1}$ and same critical
constant.

The first (and central) part of the proof is to establish that for $t>0$
fixed, $h_{t}$ is Lipschitz continuous (hence continuous). Note that the
second point of Proposition 1.5.3 also follows from that assertion. We omit
this technical (and central) aspect and refer to [93].

We now apply the discrete weak KAM theorem 1.2.1 which states that for all
$n\in\mathbb{N}$, there exists a unique constant $c_{n}$ and a function
$u_{n}:M\to\mathbb{R}$ such that $u_{n}=S^{-}(2^{-n})u_{n}+c_{n}$. By using
the semigroup property. One obtains that

$u_{n}=\big{(}S^{-}(2^{-n})\big{)}^{2^{n}}u_{n}+2^{n}c_{n}=S^{-}(1)u_{n}+2^{n}c_{n}.$

It follows by the uniqueness of $c_{0}$ that for all $n\geqslant 0$,
$c_{n}=2^{-n}c_{0}$. The same argument yields that

$\forall n\geqslant 0,\ \forall t\in 2^{-n}\mathbb{N},\quad
u_{n}=S^{-}(t)u_{n}+tc_{0}.$ (1.7)

Up to adding constants to the functions $u_{n}$ we may assume they all vanish
at some point of $M$ which does not change their property of being weak KAM
solutions for $S^{-}(2^{-n})$. Moreover, as all the $u_{n},n\in\mathbb{N}$ are
in the image of $S^{-}(1)$ they form an equi–Lipschitz family of functions.
Hence by the Arzelà–Ascoli theorem, we may find an extracted sequence $k_{n}$
such that $(u_{k_{n}})_{n\in\mathbb{N}}$ converges uniformly to a function
$v$. By continuity of $u\mapsto S^{-}(t)u$ we may pass to the limit (as
$m\to+\infty$) in the equalities

$\forall m\geqslant n,\ \forall t\in 2^{-k_{n}}\mathbb{N},\quad
u_{k_{m}}=S^{-}(t)u_{k_{m}}+tc_{0},$

to obtain that $v=S^{-}(t)v+tc_{0}$ for any diadic number $t$. As diadic
numbers are dense in $[0,+\infty)$ the theorem follows again by continuity of
$t\mapsto S^{-}(t)v$. ∎

We have used at the end of the proof the following result, of which we give a
quick proof for completeness:

######  Lemma 1.5.7.

Let $u:M\to\mathbb{R}$ be a continuous function, then the function $t\mapsto
S^{-}(t)u$ is uniformly continuous.

###### Proof.

Using the semigroup property and non–expansiveness, one obtains that for all
$0\leqslant s\leqslant t$,
$\|S^{-}(t)u-S^{-}(s)u\|_{\infty}\leqslant\|S^{-}(t-s)u-u\|_{\infty}$.
Therefore, it is enough to prove continuity at $0$. Moreover, again using the
non–expansive character, one sees by an approximation argument that it is
enough to prove the result for $u$ being a Lipschitz function.

Assume therefore that $u$ is Lipschitz continuous with Lipschitz constant
$K>0$. As $L$ is superlinear, there is a constant $C>0$ such that

$\forall(x,v)\in TM,\quad L(x,v)\geqslant K\|v\|_{x}-C.$

It follows that for any absolutely continuous $\gamma:[-t,0]\to M$,

$\int_{-t}^{0}L\big{(}\gamma(s),\dot{\gamma}(s)\big{)}{\rm
d}s\geqslant\int_{-t}^{0}K\|\dot{\gamma}(s)\|_{\gamma(s)}{\rm d}s-tC\geqslant
K{\rm d}\big{(}\gamma(0),\gamma(t)\big{)}-tC.$

It follows that if $\gamma(0)=x$, recalling that $u$ is $K$–Lipschitz,

$\int_{-t}^{0}\Big{[}L\big{(}\gamma(s),\dot{\gamma}(s)\big{)}+\alpha\Big{]}{\rm
d}s+u\big{(}\gamma(-t)\big{)}-u(x)\geqslant\\\ \geqslant K{\rm
d}\big{(}\gamma(0),\gamma(t)\big{)}-tC-K{\rm
d}\big{(}\gamma(0),\gamma(t)\big{)}$

and taking an infimum on all curves, $S^{-}(t)u(x)-u(x)\geqslant-tC$.

Finally, comparing with a constant curve in the definition of the Lax–Oleinik
semigroup one finds that

$S^{-}(t)u(x)\leqslant u(x)+\int_{-t}^{0}L(x,0){\rm d}s\leqslant
u(x)+t\max_{y\in M}L(y,0).$

Those two inequalities prove the lemma. ∎

One may wonder if there is a relationship between discrete weak KAM solutions
and weak KAM solutions. It turns out that the answer is yes and it is closely
related to the autonomous aspect of our setting555The second part of the
following Theorem becomes tautological when considering time–dependent
$1$–periodic Hamiltonians as the definitions of weak KAM solutions then
coincide..

######  Theorem 1.5.8.

Let $c=h_{1}$ be the cost function. Then we have $\alpha(0)=c[0]$. Moreover a
function $u$ is a discrete weak KAM solution for $c$ if and only if it is a
weak KAM solution for $H$.

The proof heavily relies on Fathi’s Theorem [91]:

######  Theorem 1.5.9 (Fathi [91]).

Let $v:M\to\mathbb{R}$ be a continuous function. Then $t\mapsto
S^{-}(t)v+t\alpha(0)$ uniformly converges to a weak KAM solution, for $H$, as
$t\to+\infty$.

###### Proof of Theorem 1.5.8.

It is clear that if $u$ is a weak KAM solution, then
$u=S^{-}(1)u+\alpha(0)=T^{-}u+\alpha(0)$. It follows that $\alpha(0)=c[0]$
because of the uniqueness in Theorem 1.2.1. Moreover, any weak KAM solution is
a discrete weak KAM solution.

Let now $v$ be a discrete weak KAM solution. It follows that
$v=S^{-}(1)v+\alpha(0)$ and then that

$\forall n\in\mathbb{N},\quad v=S^{-}(n)v+n\alpha(0).$

By Fathi’s theorem, there exists a weak KAM solution $\tilde{v}$ such that
$S^{-}(t)v+t\alpha(0)\to\tilde{v}$ as $t\to+\infty$. It then follows that
$v=\tilde{v}$ is a weak KAM solution. ∎

If $\alpha\in\mathbb{R}$, denote by $\mathcal{S}^{\prime}_{\alpha}$ the set of
subsolutions to (SHJ), or equivalently functions verifying (1.6). Note that
$u\in\mathcal{S}^{\prime}_{\alpha}$ if and only if $t\mapsto
S^{-}(t)u+t\alpha$ is non–decreasing. Finally, $\mathcal{S}^{\prime}$ will
denote the special set of critical subsolutions
$\mathcal{S}^{\prime}_{\alpha(0)}$. As will be seen later, unlike what happens
for weak KAM solutions, the set
$\mathcal{S}^{\prime}_{\alpha}\subset{\mathcal{S}}_{\alpha}$ is much smaller
than its discrete analogue. The first point of the next proposition is a hint
as to why. We state without proof analogous results to Proposition 1.2.6:

######  Proposition 1.5.10.

Let $\alpha\in\mathbb{R}$, the following assertions hold:

  1. (i)

The family of $\alpha$–subsolutions is equi–Lipschitz.

  2. (ii)

The set $\mathcal{S}^{\prime}_{\alpha}$ of $\alpha$–subsolutions is closed
(with respect to pointwise and uniform convergence), convex and stable by the
Lax–Oleinik semigroup:
$S^{-}(t)(\mathcal{S}^{\prime}_{\alpha})\subset\mathcal{S}^{\prime}_{\alpha}$
for all $t\geqslant 0$.

And here is the characterization of $\alpha(0)$ similar to Proposition 1.2.7:

######  Proposition 1.5.11.

The following equality holds: $\alpha(0)=\min\\{\alpha\in\mathbb{R},\ \
\mathcal{S}^{\prime}_{\alpha}\neq\varnothing\\}$.

We finish this section by discussing the continuous way of adapting the second
proof of the discrete weak KAM theorem. Indeed, if $\lambda\in(0,1)$,
considering the mapping $u\mapsto S^{-}(1)(\lambda u)$ does not make much
sense from the PDE point of view.

The function $u_{\lambda}$ constructed in the second proof of Theorem 1.2.1
satisfies $T^{-}(\lambda u_{\lambda})=u_{\lambda}$. By setting
$v_{\lambda}=\lambda u_{\lambda}$, the previous equation may be rewritten as

$v_{\lambda}-T^{-}(v_{\lambda})=(\lambda-1)u_{\lambda}=(1-\lambda^{-1})v_{\lambda}.$

We now follow the intuition that $T^{-}$stands for the time $1$ of an
evolution semigroup, hence $v_{\lambda}-T^{-}(v_{\lambda})$ can be interpreted
as a discrete derivative with respect to time.

Going back to the Hamilton–Jacobi equation, and following the previous
analysis, we are looking for a function $u:M\to\mathbb{R}$ such that

$\frac{{\rm d}}{{\rm d}t}S^{-}(t)u_{|t=0}=-\ell u,$

where we applied the change of variable $\ell=\lambda^{-1}-1$. Hence $\ell>0$
is aimed to go to $0$. Remembering that $(t,x)\mapsto S^{-}(t)u(x)$ solves the
Hamilton–Jacobi equation, we are actually looking for a function $u$ solving

$\ell u(x)+H(x,D_{x}u)=0,\quad x\in M$ ($\ell$HJ)

in the viscosity sense. The preceding equation is called the discounted
equation and $\ell$ is called the discount factor. It turns out this is
precisely the method used in [132] which is historically the first paper where
weak KAM solutions appear. In this foundational preprint, they prove the
following results, for a wider class of Hamiltonians (in particular, no
convexity is required):

######  Theorem 1.5.12.

  1. 1.

For all $\ell>0$ there exists a unique function $U_{\ell}:M\to\mathbb{R}$
which is a viscosity solution of ($\ell$HJ).

  2. 2.

The family $(\ell U_{\ell})_{\ell>0}$ is equi-bounded.

  3. 3.

The family $(U_{\ell})_{\ell>0}$ is equi-Lipschitz.

  4. 4.

Given $x_{0}\in M$ and setting $v_{\ell}=U_{\ell}-U_{\ell}(x_{0})$, it follows
that the family $(v_{\ell})_{\ell>0}$ is relatively compact.

  5. 5.

The family $(\ell U_{\ell})_{\ell>0}$ uniformly converges to the constant
function $-\alpha(0)$666In the preprint, this constant is actually denoted by
$-\overline{H}(0)$ and $\overline{H}$ is called the effective Hamiltonian. as
$\ell\to 0$ and any limit function $v_{0}$ of the family
$(v_{\ell})_{\ell>0}$, as $\ell\to 0$, is a solution of (SHJ) with right hand
side $\alpha(0)$.

####  1.5.3 The positive Lax–Oleinik semigroup

As in our discrete setting, Fathi introduced the positive Lax–Oleinik
semigroup as follows:

######  Definition 1.5.13.

Let $u:M\to\mathbb{R}$ be any continuous function, for any $t>0$ and $x\in M$
we define:

$S^{+}(t)u(x)=\sup_{\begin{subarray}{c}\gamma:[0,t]\to M\\\
\gamma(0)=x\end{subarray}}u\big{(}\gamma(t)\big{)}-\int_{0}^{t}L\big{(}\gamma(s),\dot{\gamma}(s)\big{)}{\rm
d}s=\sup_{y\in M}u(y)-h_{t}(x,y),$

where the supremum is taken amongst absolutely continuous curves.

Once again, this semigroup has a natural PDE interpretation. Let us introduce
the Hamiltonian: $\widecheck{H}:(x,p)\mapsto H(x,-p)$. One verifies that the
associated Lagrangian is given by

$\forall(x,v)\in TM,\quad\widecheck{L}(x,v)=\sup_{p\in
T^{*}_{x}M}p(v)-\widecheck{H}(x,p)=\sup_{p\in
T^{*}_{x}M}p(-v)-H(x,p)=L(x,-v).$

Therefore the positive semigroup is written as follows:

$S^{+}(t)u(x)=-\inf_{\begin{subarray}{c}\gamma:[0,t]\to M\\\
\gamma(0)=x\end{subarray}}-u\big{(}\gamma(t)\big{)}+\int_{0}^{t}L\big{(}\gamma(s),\dot{\gamma}(s)\big{)}{\rm
d}s\\\ =-\bigg{[}\inf_{\begin{subarray}{c}\check{\gamma}:[-t,0]\to M\\\
\check{\gamma}(0)=x\end{subarray}}-u\big{(}\check{\gamma}(t)\big{)}+\int_{-t}^{0}\widecheck{L}\big{(}\check{\gamma}(s),\dot{\check{\gamma}}(s)\big{)}{\rm
d}s\bigg{]}=-\widecheck{S}(t)(-u)(x),$

where $\widecheck{S}$ denotes the Lax–Oleinik semigroup associated to
$\widecheck{H}$. Hence the function $(t,x)\mapsto-S^{+}(t)u(x)$ solves an
evolutionary Hamilton–Jacobi equation with Hamiltonian $\widecheck{H}$ and
initial data $-u$.

As the Hamiltonian $\widecheck{H}$ is also Tonelli, it is automatic that the
positive semigroup $S^{+}$ has the same properties as $S^{-}$ stated in
Proposition 1.5.3, that we do not recall here. As for the discrete case, a
positive weak KAM theorem follows:

######  Theorem 1.5.14 (positive weak KAM).

The critical constant $\alpha(0)$ is the only one for which the equation
$u=S^{+}(t)u-t\alpha(0)$, for all $t>0$, admits solutions $u:M\to\mathbb{R}$.
Moreover the constant $\alpha(0)$ has the following caracterization:

$\forall
v\in{\mathcal{B}}(M,\mathbb{R}),\quad\frac{S^{+}(t)v}{t}\underset{t\to+\infty}{\longrightarrow}\alpha(0),$

and the convergence is uniform.

####  1.5.4 Strict subsolutions, Aubry sets

As is expected, given a constant $\alpha\in\mathbb{R}$, a function
$u:M\to\mathbb{R}$ is an $\alpha$–subsolution
($u\in\mathcal{S}^{\prime}_{\alpha}$) if

$\forall t>0,\ \forall\gamma:[-t,0]\to M,\quad
u\big{(}\gamma(0)\big{)}-u\big{(}\gamma(-t)\big{)}\leqslant\int_{-t}^{0}\Big{[}L\big{(}\gamma(s),\dot{\gamma}(s)\big{)}+\alpha\Big{]}{\rm
d}s,$

where $\gamma$ ranges in the set of absolutely continuous curves. It can be
established that subsolutions are automatically Lipschitz hence differentiable
almost everywhere.

The terminology comes from the fact that subsolutions are indeed viscosity
subsolutions to the critical stationary Hamilton–Jacobi equation (SHJ). Whence
the following characterizations of subsolutions hold:

######  Proposition 1.5.15.

Let $u:M\to\mathbb{R}$ and $\alpha\in\mathbb{R}$ be a constant. The following
are equivalent:

  1. 1.

$u\in\mathcal{S}^{\prime}_{\alpha}$ is an $\alpha$–subsolution;

  2. 2.

the family of functions $\big{(}S^{-}(t)u+t\alpha\big{)}_{t\geqslant 0}$ is
non–decreasing;

  3. 3.

the family of functions $\big{(}S^{+}(t)u-t\alpha\big{)}_{t\geqslant 0}$ is
non–increasing;

  4. 4.

the function $u$ verifies $H(x,D_{x}u)\leqslant\alpha$ in the viscosity sense;

  5. 5.

the function $u$ verifies $H(x,D_{x}u)\leqslant\alpha$ for almost every $x\in
M$ (more precisely at every $x\in M$ such that $u$ is differentiable at $x$).

We now focus on the particular case of critical subsolutions. The idea is
that, because the constant $\alpha(0)$ is a threshold between a world with
subsolutions and a world without, there is no critical subsolution where the
inequality $H(x,D_{x}u)\leqslant\alpha(0)$ is everywhere strict. More
precisely, the obstruction to having strict inequalities is concentrated on a
subset of $M$: the projected Aubry set. This set was introduced for twist maps
(that are a particular discrete setting) by Aubry, Le Daeron and Mather [18,
138, 148]. For Hamiltonian systems, it was later on introduced by Mather
([143]) by dynamical systems means. The present interpretation, using critical
subsolutions, is due to Fathi ([93, 89, 88, 90]).

A fundamental result on subsolutions is due to Fathi and Siconolfi [99, 100]
(for $C^{1}$ subsolutions) and was then improved by Bernard ([33]):

######  Theorem 1.5.16.

There exists a function $u_{0}:M\to\mathbb{R}$ that is $C^{1,1}$ ($C^{1}$ with
Lipschitz differential) and is a critical subsolution
($u_{0}\in\mathcal{S}^{\prime}_{\alpha_{0}}$). Moreover it verifies the
following fundamental property:

if for some $x\in M$, $H(x,D_{x}u_{0})=\alpha(0)$ then if
$u\in\mathcal{S}^{\prime}_{\alpha(0)}$ is any other critical subsolution, $u$
is differentiable at $x$ and $D_{x}u=D_{x}u_{0}$. Hence
$H(x,D_{x}u)=\alpha(0)$.

The function $u_{0}$ above is a $C^{1,1}$ strict subsolution. The set

${\mathcal{A}}=\\{x\in M,\ \ H(x,D_{x}u_{0})=\alpha(0)\\},$

is called the projected Aubry set. It is non–empty (otherwise $u_{0}$ would be
a subsolution for a constant less that $\alpha(0)$). It is also compact. The
Aubry set is ${\mathcal{A}}^{*}=\\{(x,D_{x}u_{0}),\ \
x\in{\mathcal{A}}\\}\subset T^{*}M$. Of course, ${\mathcal{A}}^{*}\subset
H^{-1}(\\{\alpha(0)\\})$. This is actually a deep fact. It implies a Theorem
of Carneiro ([62]) which we come back to later. It may seem at first glance
that those sets depend on $u_{0}$ but it is not the case. Indeed,
${\mathcal{A}}$ is a set of points where all subsolutions $u$ are
differentiable777This actually suffices to characterize the projected Aubry
set as proven in [99]. and $H(x,D_{x}u)=\alpha(0)$. The Aubry set in $TM$ is
then naturally ${\mathcal{A}}^{\prime}={\mathcal{L}}^{-1}({\mathcal{A}}^{*})$.

As we saw in the discrete setting, points of the projected Aubry set come in
whole sequences, giving rise to the Aubry set (subset of $X^{\mathbb{Z}}$).
Moreover, this Aubry set is invariant by the shift on $X^{\mathbb{Z}}$.
Similar phenomena arise in the classical setting, the dynamical systems here
being those of $\varphi_{L}$ the Euler–Lagrange flow and $\varphi_{H}$ the
Hamiltonian flow:

######  Theorem 1.5.17.

The Aubry set ${\mathcal{A}}^{*}$ is invariant by $\varphi_{H}$ meaning that
for all $t\in\mathbb{R}$,
$\varphi_{H}^{t}({\mathcal{A}}^{*})={\mathcal{A}}^{*}$. Equivalently, the
Aubry set ${\mathcal{A}}^{\prime}$ is invariant by $\varphi_{L}$ meaning that
for all $t\in\mathbb{R}$,
$\varphi_{L}^{t}({\mathcal{A}}^{\prime})={\mathcal{A}}^{\prime}$.

It follows that if $(x,p)\in{\mathcal{A}}^{*}$,
$u\in\mathcal{S}^{\prime}_{\alpha(0)}$ is a critical subsolution ant
$t\in\mathbb{R}$, then $\big{(}x(t),p(t)\big{)}=\big{(}x(t),D_{x(t)}u\big{)}$
where we set $\big{(}x(t),p(t)\big{)}=\varphi_{H}^{t}(x,p)$. In particular, we
stress that $u$ is automatically differentiable at $x(t)$ for all
$t\in\mathbb{R}$.

The curve $\big{(}x(t),\dot{x}(t)\big{)}_{t\in\mathbb{R}}$ is a trajectory of
the Euler–Lagrange flow. Moreover, its projection on $M$ calibrates $u$ in the
sense that

$\forall s<t,\quad
u\big{(}x(t)\big{)}-u\big{(}x(s)\big{)}=\int_{s}^{t}L\big{(}x(\sigma),\dot{x}(\sigma)\big{)}{\rm
d}\sigma+(t-s)\alpha(0).$

In particular, recalling that $u$ verifies (1.6), it follows that

$\displaystyle\forall t>0,\quad$ $\displaystyle
S^{-}(t)u(x)=u\big{(}x(-t)\big{)}+\int_{-t}^{0}L\big{(}x(\sigma),\dot{x}(\sigma)\big{)}{\rm
d}\sigma+t\alpha(0)=u(x)+t\alpha(0);$ $\displaystyle
S^{+}(t)u(x)=u\big{(}x(t)\big{)}-\int_{0}^{t}L\big{(}x(\sigma),\dot{x}(\sigma)\big{)}{\rm
d}\sigma-t\alpha(0)=u(x)-t\alpha(0).$

From this follows the analogue of Corollary 1.4.10:

######  Corollary 1.5.18.

Let $x\in M$. The following assertions are equivalent:

  1. (i)

$x\in{\mathcal{A}}$,

  2. (ii)

$\forall u\in\SS^{\prime}_{\alpha(0)},\ \forall t>0,\quad
u(x)=S^{-}(t)u(x)+t\alpha(0)$,

  3. (iii)

$\forall u\in\SS^{\prime}_{\alpha(0)},\ \forall t>0,\quad
u(x)=S^{+}(t)u(x)-t\alpha(0)$.

We end this section with a new description of the relation between the Aubry
set for the Hamiltonian and the Aubry set for the time–$1$ action functional
$h_{1}$. For the sake of clarity, we denote them respectively
${\mathcal{A}}_{H}$ and ${\mathcal{A}}_{h_{1}}$.

######  Theorem 1.5.19.

The equality ${\mathcal{A}}_{H}={\mathcal{A}}_{h_{1}}$ holds.

###### Proof.

Recall that if $n>0$, thanks to the choice of cost function, the Lax–Oleinik
semigroups are linked by the equalities $T^{-n}=S^{-}(n)$ and
$T^{+n}=S^{+}(n)$. Moreover, we have seen that both critical constants for $H$
and $h_{1}$ coincide. It follows that classical subsolutions are discrete
subsolutions,
$\mathcal{S}^{\prime}_{\alpha(0)}\subset{\mathcal{S}}_{\alpha(0)}$ (the
inclusion being very often strict). Indeed,
$u\in\mathcal{S}^{\prime}_{\alpha(0)}$ if and only if $t\mapsto
S^{-}(t)u+t\alpha(0)$ is non–decreasing for $t\in[0,+\infty)$ while
$u\in{\mathcal{S}}_{\alpha(0)}$ if and only if the sequence $n\mapsto
S^{-}(n)u+n\alpha(0)$ is non–decreasing for $n\geqslant 0$.

Let now $x\in{\mathcal{A}}_{h_{1}}$ and
$u\in\mathcal{S}^{\prime}_{\alpha(0)}$. We deduce from the preceding
discussion and Corollary 1.4.10 that the sequence $n\mapsto
S^{-}(n)u(x)+n\alpha(0)$ is constant. As the familly $t\mapsto
S^{-}(t)u+t\alpha(0)$ is non–decreasing, it has to be constant. This being
true for any classical subsolution, we deduce that $x\in{\mathcal{A}}_{H}$ by
Corollary 1.5.18. We have established that
${\mathcal{A}}_{h_{1}}\subset{\mathcal{A}}_{H}$.

Let then $x\in{\mathcal{A}}_{H}$ and $u\in{\mathcal{S}}_{\alpha(0)}$ be a
discrete subsolution. We set
$u_{-}=\lim\limits_{n\to+\infty}S^{-}(n)u+n\alpha(0)$ and
$u_{+}=\lim\limits_{n\to+\infty}S^{+}(n)u-n\alpha(0)$ that both exist (the
sequences are monotonous). They are respectively a negative and positive weak
KAM solution and verify $u_{+}\leqslant u\leqslant u_{-}$. Finally, let us set
$v=\lim\limits_{t\to+\infty}S^{-}(t)u_{+}+t\alpha(0)$ that is a negative weak
KAM solution. As classical and discrete weak KAM solutions coincide, we know
from Corollary 1.5.18 that $u_{+}(x)=S^{-}(t)u_{+}(x)+t\alpha(0)$ for all
$t>0$, hence $u_{+}(x)=v(x)$.

We will prove that $v=u_{-}$. Let $\varepsilon>0$ and $N>0$ such that
$\|S^{+}(n)u-n\alpha(0)-u_{+}\|<\varepsilon$ for all $n\geqslant N$. By
application of Proposition 1.3.6 and monotonicity of the semigroups, one
establishes that $S^{-}(n)\circ S^{+}(n)u\geqslant u$. By taking $n>N$ and
using the monotonicity of $S^{-}$ again, it follows that

$S^{-}(n)u_{+}+n\alpha(0)\geqslant S^{-}(n)\circ
S^{+}(n)u-\varepsilon\geqslant u-\varepsilon.$

Then if $m>0$ we obtain that $S^{-}(n+m)u_{+}+(n+m)\alpha(0)\geqslant
S^{-}(m)u+m\alpha(0)-\varepsilon$. Finally letting $m\to+\infty$ it follows
that $v\geqslant u_{-}-\varepsilon$, and as this is true for all
$\varepsilon>0$ we obtain that $v\geqslant u_{-}$.

The reverse inequality is easier: as $u_{+}\leqslant u_{-}$ then

$\forall n>0,\quad S^{-}(n)u_{+}+n\alpha(0)\leqslant
S^{-}(n)u_{-}+n\alpha(0)=u_{-},$

and we conclude by letting $n\to+\infty$.

Finally, it follows that $u_{+}(x)=v(x)=u_{-}(x)$ and as for $n>0$,

$u_{+}\leqslant u\leqslant S^{-}(n)u+n\alpha(0)\leqslant u_{-},$

we also have $u(x)=S^{-}(n)u(x)+n\alpha(0)=u_{-}(x)$. Thus
$x\in{\mathcal{A}}_{h_{1}}$ and the proof is complete. ∎

In the preceding proof the functions $u_{-}$ and $u_{+}$ form what is called a
conjugate pair (see [88] for the classical definition and [40] for definitions
in a discrete setting), a notion that will reappear in this text (as in Remark
2.1.4).

##  Chapter 2 More (dynamical) characterizations of the Aubry sets

So far, we have constructed several different Aubry sets and have started to
study the way they are related. Even though their initial definition stems
from one particular subsolution, we already began to understand that, in the
end, they entail informations concerning all subsolutions. In fact they depend
on the cost function $c$ only, and not on the particular subsolution $u_{0}$.
In this section we push further the understanding of the meaning of those
Aubry sets.

Most results of this Chapter are to be found in [174] by the author for the
most abstract and general part. When it comes to the more regular setting of
costs defined on a compact manifold first results are written in [173] by the
author, then generalized in [46] with Bernard.

###  2.1 The Peierls Barrier

The aim of this paragraph is to give a characterization of the Aubry set in
terms of action along long chains of points. This characterization builds on
the following notion:

######  Definition 2.1.1.

If $n>0$ is a positive integer and $(x,y)\in X\times X$, let

$c_{n}(x,y)=\inf\left\\{\sum\limits_{i=0}^{n-1}c(x_{i},x_{i+1}),\quad(x_{0},x_{1},\cdots,x_{n-1},x_{n})\in
X^{n+1},\ x_{0}=x,\ x_{n}=y\right\\}.$

The Peierls barrier is the function $h:X\times X\to X$ defined by

$\forall(x,y)\in X\times X,\quad
h(x,y)=\liminf\limits_{n\to+\infty}c_{n}(x,y)+n{c[0]}.$

It follows from the previous Definition that if
$u\in{\mathcal{B}}(X,\mathbb{R})$ is a bounded function, then
$T^{-n}u(x)=\inf\limits_{y\in X}u(y)+c_{n}(y,x)$.

The name Peierls barrier appears in Aubry and Le Daeron’s work [18] in the
context of Conservative Twist Maps of the Annulus (see the last Chapter of
this text). It associates to each rotation number a real number that vanishes
if and only if there is an invariant circle with this rotation number (all
those notions are defined in the last Chapter). Hence it is a barrier to the
existence of invariant circles. Mather then introduced variations and studied
properties of this barrier, still in the context of Twist Maps in [137]. He
went on to generalize the Peierls barrier to higher dimensional Lagrangian
settings in [145]. The Definition just presented is analogous to what is done
in the latter reference.

Let us start by giving some properties of this new object:

######  Proposition 2.1.2.

  1. (i)

$h$ is well defined and continuous;

  2. (ii)

for all $u\in{\mathcal{S}}$ and $(x,y)\in X$, $u(y)-u(x)\leqslant h(x,y)$, in
particular for all $x\in X$, $h(x,x)\geqslant 0$;

  3. (iii)

for all $x,y,z\in X$ and integer $n>0$ the following inequalities hold:

$h(x,y)\leqslant h(x,z)+c_{n}(z,y)+n{c[0]};\\\ h(x,y)\leqslant
c_{n}(x,z)+n{c[0]}+h(z,y);\\\ h(x,y)\leqslant h(x,z)+h(z,y);$ (2.1)

  4. (iv)

for all $x\in X$, the function $h_{x}=h(x,\cdot)$ is a weak KAM solution and
the function $h^{x}=-h(\cdot,x)$ is a positive weak KAM solution.

###### Proof.

Let $\omega$ denote a modulus of continuity for $c$: a non–decreasing function
from $[0,+\infty)$ to itself that is continuous at $0$, with $\omega(0)=0$,
such that

$\forall(x,y,x^{\prime},y^{\prime})\in
X^{4},\quad|c(x,y)-c(x^{\prime},y^{\prime})|\leqslant\omega\big{(}d(x,x^{\prime})+d(y,y^{\prime})\big{)}.$

Let $x,x^{\prime},y,y^{\prime}$ be points, $n$ an integer, and
$x=x_{0},x_{1},\cdots,x_{n-1},x_{n}=y$ such that
$c_{n}(x,y)=\sum\limits_{i=0}^{n-1}c(x_{i},x_{i+1})$ (they exist by
compactness of $X$ and continuity of the maps). Then one gets by definition
that

$c_{n}(x^{\prime},y^{\prime})-c_{n}(x,y)\\\ \leqslant
c(x^{\prime},x_{1})+\sum\limits_{i=1}^{n-2}c(x_{i},x_{i+1})+c(x_{n-1},y^{\prime})-\sum\limits_{i=0}^{n-1}c(x_{i},x_{i+1})\\\
\leqslant\omega\big{(}d(x,x^{\prime})+d(y,y^{\prime})\big{)}.$

We get the same inequality for $c_{n}(x,y)-c_{n}(x^{\prime},y^{\prime})$ by
the same argument, which proves that the $c_{n}$ are equicontinuous. Moreover,
one checks readily that $c_{n+1}(x,y)=T^{-n}c(x,\cdot)(y)$. Therefore,
combined with the first point of Remark 1.2.4, this implies that the family on
functions $c_{n}+n{c[0]}$ is uniformly bounded. This proves (i).

We have already seen in Remark 1.4.3-(iii) that if $u\in{\mathcal{S}}$ and
$x,y$ are points, then for any integer $n>0$, $u(y)-u(x)\leqslant
c_{n}(x,y)+n{c[0]}$. Taking a liminf yields (ii).

One has by definition that if $m,n$ are integers and $x,y$ are points, then
$c_{n+m}(x,y)=\inf\limits_{Z\in X}c_{m}(x,Z)+c_{n}(Z,y)$. Hence if $z$ is a
third point, then

$c_{n+m}(x,y)+(n+m){c[0]}\leqslant c_{m}(x,z)+m{c[0]}+c_{n}(z,y)+n{c[0]}.$

Letting $m\to+\infty$ and taking liminf leads to the first inequality of
(iii), $n\to+\infty$ and taking liminf to the second one, and both at the same
time to the third.

Let $x\in X$. The previous inequalities applied to $n=1$ show that the
functions $h_{x}$ and $h^{x}$ are subsolutions. Let us prove (iv) for $h_{x}$,
the rest being similar. Let $y\in X$ and $k_{n}$ be an extraction such that
$h_{x}(y)=\lim\limits_{n\to+\infty}c_{k_{n}+1}(x,y)+(k_{n}+1){c[0]}$. For each
$n$, there exists $x_{n}\in X$ such that
$c_{k_{n}+1}(x,y)+(k_{n}+1){c[0]}=c_{k_{n}}(x,x_{n})+c(x_{n},y)+(k_{n}+1){c[0]}$
and taking a further extraction, one may assume that $x_{n}\to\tilde{x}$ for
some $\tilde{x}\in X$. Taking the limit and by equicontinuity, one concludes
that

$h_{x}(y)=\lim\limits_{n\to+\infty}c_{k_{n}}(x,x_{n})+k_{n}{c[0]}+c(x_{n},y)+{c[0]}\geqslant
h(x,\tilde{x})+c(\tilde{x},y)+{c[0]}.$

The function $h_{x}$ being a subsolution, the last inequality must be an
equality and it follows that $h_{x}(y)=T^{-}h_{x}(y)+{c[0]}$ which is what was
to be proven. ∎

We can actually strengthen (ii) in the previous Proposition as follows:

######  Proposition 2.1.3.

Let $u\in{\mathcal{S}}$ and $m,n$ two non–negative integers, then

$\forall(x,y)\in X\times X,\quad T^{-n}u(y)-T^{+m}u(x)+(n+m){c[0]}\leqslant
h(x,y).$

###### Proof.

Let $m,n$ be any non–negative integers and $x=x_{-m},\cdots,x_{n}=y$ be a
chain of points. Then by definition of the Lax–Oleinik semigroups, one gets

$T^{-n}u(y)-T^{+m}u(x)\leqslant\sum_{i=0}^{n-1}c(x_{i},x_{i+1})+u(x_{0})-u(x_{0})+\sum_{i=-m}^{-1}c(x_{i},x_{i+1})=\sum_{i=-m}^{n-1}c(x_{i},x_{i+1}).$

This being true for all chains of points it follows that

$T^{-n}u(y)-T^{+m}u(x)+(n+m){c[0]}\leqslant c_{n+m}(x,y)+(n+m){c[0]}.$

Note that the left hand side is now non–decreasing in both $n$ and $m$ (as
$u\in{\mathcal{S}}$). Therefore to show the statement, we only have to show
that the limit when $n,m\to+\infty$ verifies the same inequality. Let
$n_{k},m_{k}$ be two diverging increasing sequences such that
$c_{n_{k}+m_{k}}(x,y)+(n_{k}+m_{k}){c[0]}\to h(x,y)$. Then one concludes that

$T^{-n}u(y)-T^{+m}u(x)+(n+m){c[0]}\leqslant\\\
\leqslant\lim_{k\to+\infty}T^{-n_{k}}u(y)-T^{+m_{k}}u(x)+(n_{k}+m_{k}){c[0]}\\\
\leqslant\lim_{k\to+\infty}c_{n_{k}+m_{k}}(x,y)+(n_{k}+m_{k}){c[0]}=h(x,y),$

which is the result. ∎

######  Remark 2.1.4.

The previous proposition can actually be stated in a more optimal way by
introducing another notion. Given a subsolution $u\in{\mathcal{S}}$, as noted,
the sequences $T^{-n}u+n{c[0]}$ and $T^{+n}u-n{c[0]}$ converge respectively to
functions $u^{-}$ and $u^{+}$ which are respectively a negative and a positive
weak KAM solution. The result then states that $u^{-}(y)-u^{+}(x)\leqslant
h(x,y)$. Such pairs are called conjugate pairs, they coincide on
${\mathcal{A}}$ (Corollary 1.4.10) and this last point characterizes the pair
$(u^{-},u^{+})$ (see proposition 2.2.5 in the next paragraph).

We are now ready to establish a characterization of the Aubry sets much easier
to handle:

######  Theorem 2.1.5.

The following equalities hold:

${\mathcal{A}}=\\{x\in X,\quad h(x,x)=0\\},$
$\widehat{\mathcal{A}}=\\{(x,y)\in X\times X,\quad c(x,y)+{c[0]}+h(y,x)=0\\}.$

###### Beginning of the Proof.

We will only prove two inclusions for now. The other ones will arrive after an
intermediate Proposition 2.1.6.

Let us call $A_{1}\subset X$, $A_{2}\subset X\times X$ the sets appearing in
the right hand side of the statement. If $u\in{\mathcal{S}}$ and $h(x,x)=0$,
by the previous Proposition both sequences
$(T^{-n}u(x)+n{c[0]})_{n\in\mathbb{N}}$ and
$(T^{+n}u(x)-n{c[0]})_{n\in\mathbb{N}}$ are constantly equal to $u(x)$. This
implies that $x\in{\mathcal{A}}$ (see Corollary 1.4.10) and
$A_{1}\subset{\mathcal{A}}$.

Assume $(x,y)\in A_{2}$. Then for any $u\in{\mathcal{S}}$, summing up the
inequalities $u(y)-u(x)\leqslant c(x,y)+{c[0]}$ and $u(x)-u(y)\leqslant
h(y,x)$ brings that $0\leqslant c(x,y)+{c[0]}+h(y,x)$. As this is an equality,
both inequalities were equalities and $(x,y)\in\widehat{\mathcal{A}}$. So
$A_{2}\subset\widehat{\mathcal{A}}$.

∎

In order to obtain the reverse inclusions, we will use the following:

######  Proposition 2.1.6.

Let $x\in X$, then $T^{+n}h_{x}(x)-n{c[0]}\to 0$ as $n\to+\infty$.

###### Proof.

Recall $T^{+n}h_{x}(x)-n{c[0]}$ is a decreasing sequence
($h_{x}\in{\mathcal{S}}$). Moreover, as
$h_{x}(y)-T^{+n}h_{x}(x)+n{c[0]}\leqslant h(x,y)$, we see that
$T^{+n}h_{x}(x)-n{c[0]}\geqslant 0$ for all $n>0$ and the limit meets the same
property.

For each $n>0$, let $x=x_{0}^{n},\cdots,x_{n}^{n}$ verify
$T^{+n}h_{x}(x)=h_{x}(x_{n}^{n})-\sum\limits_{i=0}^{n-1}c(x_{i}^{n},x_{i+1}^{n})$.
Up to extracting, assume $x_{n_{k}}^{n_{k}}\to y$ for some $y$, passing to the
limit, by definition of $h$ we obtain that:

$\lim_{k\to+\infty}T^{+n_{k}}h_{x}(x)-n_{k}{c[0]}\leqslant h_{x}(y)-h(x,y),$

which proves the Proposition. ∎

###### End of the Proof of Theorem 2.1.5.

Let $x\in{\mathcal{A}}$, then $T^{+n}h_{x}(x)-n{c[0]}$ is constant, hence it
is $0$, so $h(x,x)=0$.

Let $(x,y)\in\widehat{\mathcal{A}}$, then $h_{y}(y)-h_{y}(x)=c(x,y)+{c[0]}$.
But as $y\in{\mathcal{A}}$, $h(y,y)=0$ hence $c(x,y)+{c[0]}+h(y,x)=0$. ∎

Theorem 2.1.5 gives a more concrete characterization of the Aubry set. Pairs
$(x,y)$ in the $2$–Aubry set are starting pairs of arbitrarily long loops of
points with arbitrarily small cost (for the cost $c+{c[0]}$).

As a final byproduct of the preceding proofs we obtain:

######  Proposition 2.1.7.

If $(x,y)\in\widehat{\mathcal{A}}$ then $c(x,y)+c[0]=h(x,y)$.

More generally, if $(x_{n})_{n\in\mathbb{Z}}\in\widetilde{\mathcal{A}}$, then

$\forall m<n,\quad\sum_{k=m}^{n-1}c(x_{k},x_{k+1})+(n-m)c[0]=h(x_{m},x_{n}).$

###### Proof.

As the function $h_{x}$ is a critical subsolution we obtain that if
$(x,y)\in\widehat{\mathcal{A}}$, $h_{x}(y)-h_{x}(x)=c(x,y)+c[0]$. As
$h(x,x)=0$ we get $h(x,y)=c(x,y)+c[0]$.

For the second equality, we sum the equalities
$h_{x_{m}}(x_{k+1})-h_{x_{m}}(x_{k})=c(x_{k},x_{k+1})+c[0]$ to obtain

$h(x_{m},x_{n})=h_{x_{m}}(x_{n})-h_{x_{m}}(x_{m})=\sum_{k=m}^{n-1}c(x_{k},x_{k+1})+(n-m)c[0].$

∎

###  2.2 Examples of points in the Aubry sets

We will now be more specific about the type of points that belong to the Aubry
set. Most of them appear as limit points of minimizing chains. The following
lemma is most useful:

######  Lemma 2.2.1.

Let $u:X\to\mathbb{R}$ be a weak KAM solution, then for all $x\in X$ there
exists an infinite sequence $(x_{-n})_{n\geqslant 0}$ with $x_{0}=x$ such that

$\forall n\in\mathbb{N},\quad
u(x)=u(x_{-n})+\sum_{i=-n}^{-1}c(x_{i},x_{i+1})+n{c[0]}.$ (2.2)

We call such sequences, calibrating sequences for $u$.

###### Proof.

It is an immediate consequence of successive applications of the fact that for
all $x\in X$ there is $y\in X$ such that $u(x)=u(y)+c(y,x)+{c[0]}$. ∎

######  Proposition 2.2.2.

Let $u:X\to\mathbb{R}$ be a weak KAM solution and $(x_{n})_{n\geqslant 0}$ a
sequence given by Lemma 2.2.1, then $\alpha\big{(}(x_{-n})_{n\geqslant
0}\big{)}\subset{\mathcal{A}}$. If we set
$(\xi_{-n})=\big{(}(x_{-n-1},x_{-n})\big{)}_{n\geqslant 0}$ the sequence of
pairs of successive points of $(x_{-n})_{n\geqslant 0}$, then
$\alpha\big{(}(\xi_{-n})_{n\geqslant 0}\big{)}\subset\widehat{\mathcal{A}}$.
Here $\alpha$ denotes the $\alpha$–limit sets, that is limits of converging
subsequences $x_{-n_{k}}$ with $n_{k}\to+\infty$.

###### Proof.

Let $y\in\alpha\big{(}(x_{-n})_{n\geqslant 0}\big{)}$. There exists an
extraction $n_{k}\to+\infty$ such that $x_{-n_{k}}\to y$. Up to taking a
further extraction if necessary, we may assume that $n_{k+1}-n_{k}\to+\infty$
is an increasing sequence as well.

It follows from (2.2) that

$\forall k\in\mathbb{N},\quad
u(x_{-n_{k}})-u(x_{-n_{k+1}})=\sum_{i=-n_{k+1}}^{-n_{k}-1}c(x_{i},x_{i+1})+(n_{k+1}-n_{k}){c[0]}.$
(2.3)

Passing to the limit and by continuity of the functions at stake imply that
$u(y)-u(y)=0\geqslant h(y,y)$, hence $h(y,y)=0$.

The same proof holds for the second part of the proposition. Late us take an
extraction $n_{k}\to+\infty$ such that $\xi_{-n_{k}}$ converges to
$(y,y^{\prime})$ and such that $n_{k+1}-n_{k}\to+\infty$ is an increasing
sequence as well. Passing to the limit in (2.3), we get that $0\geqslant
c(y,y^{\prime})+{c[0]}+h(y^{\prime},y)$. This means
$(y,y^{\prime})\in\widehat{\mathcal{A}}$. ∎

######  Remark 2.2.3.

The same results (with same proof) hold for $\omega$–limit sets of analogous
sequences $(x_{n})_{n\geqslant 0}$ for positive weak KAM solutions.

If $u$ is a subsolution, the same results also hold for $\alpha$ and $\omega$
limit sets of elements of $\widetilde{\mathcal{A}}_{u}$.

We may now state a fundamental property of the Aubry set:

######  Theorem 2.2.4.

Let $u$ and $v$ be respectively a weak KAM solution and a subsolution such
that $u_{|{\mathcal{A}}}\geqslant v_{|{\mathcal{A}}}$, then $u\geqslant v$.

Let $u$ and $v$ be two weak KAM solutions such that
$u_{|{\mathcal{A}}}=v_{|{\mathcal{A}}}$, then $u=v$. We say ${\mathcal{A}}$ is
a uniqueness set for the critical equation.

###### Proof.

Let $x_{0}\in X$ and let $(x_{-n})_{n\geqslant 0}$ be a calibrating sequence
for $u$ (see (2.2)). It then comes

$\displaystyle\forall n\in\mathbb{N},\quad$ $\displaystyle
u(x_{0})-u(x_{-n})=\sum_{i=-n}^{-1}c(x_{i},x_{i+1})+n{c[0]},$ $\displaystyle
v(x_{0})-v(x_{-n})\leqslant\sum_{i=-n}^{-1}c(x_{i},x_{i+1})+n{c[0]}.$

Subtracting yields $u(x_{0})-v(x_{0})\geqslant u(x_{-n})-v(x_{-n})$. As
limiting points of the sequence $(x_{-n})_{n\geqslant 0}$ are in the projected
Aubry set ${\mathcal{A}}$, taking a suitable converging subsequence and
passing to the limit brings $u(x_{0})-v(x_{0})\geqslant 0$ since for $u$ and
$v$, the same inequality holds on ${\mathcal{A}}$. This proves the first
result as $x_{0}$ is arbitrary.

In the second case, by symmetry, the opposite inequality holds, and the result
follows as $x_{0}$ was taken arbitrarily.

∎

Actually, a reciprocal statement can be proven. Being a subsolution on the
Aubry set is the only obstruction to the existence of a weak KAM solution with
prescribed values on ${\mathcal{A}}$.

######  Proposition 2.2.5.

Let $f:{\mathcal{A}}\to\mathbb{R}$ be a function such that $f(y)-f(x)\leqslant
h(x,y)$ for all $x$ and $y$ in ${\mathcal{A}}$. Then there exists a weak KAM
solution $u$ such that $u_{|{\mathcal{A}}}=f$.

###### Proof.

Let us check that the function $u$ defined by
$u(x)=\inf\limits_{y\in{\mathcal{A}}}f(y)+h(y,x)$ satisfies the requirements.
Let $x\in{\mathcal{A}}$. Then $f(y)+h(y,x)\geqslant f(x)-h(y,x)+h(y,x)$ by
hypothesis. So $u(x)=f(x)$, by taking $y=x$ in the definition of $u$. The fact
that $u$ is a weak KAM solution is a consequence of the general fact that an
infimum of weak KAM solutions is a weak KAM solution applied to the family of
functions $f(y)+h_{y}$. This fact is proved in the next lemma. ∎

######  Lemma 2.2.6.

Let $(u_{\alpha})_{\alpha\in A}$ be a family of subsolutions. Set
$\overline{u}=\sup\limits_{\alpha\in A}u_{\alpha}$ and
$\underline{u}=\inf\limits_{\alpha\in A}u_{\alpha}$. Then given that those
functions are well defined, the following assertions hold:

  1. (i)

The functions $\overline{u}$ and $\underline{u}$ are subsolutions.

  2. (ii)

If all the $u_{\alpha}$ are weak KAM solutions then $\underline{u}$ is a weak
KAM solution.

###### Proof.

By definition,

$T^{-}\underline{u}(x)=\inf_{y\in X}\inf_{\alpha\in A}u_{\alpha}(y)+c(y,x)\\\
=\inf_{\alpha\in A}\inf_{y\in X}u_{\alpha}(y)+c(y,x)\\\ =\inf_{\alpha\in
A}T^{-}u_{\alpha}(x)\geqslant\inf_{\alpha\in
A}u_{\alpha}(x)-{c[0]}=\underline{u}(x)-{c[0]}.$

It was used that two infimums commute and that the $u_{\alpha}$ are
subsolutions. Hence $\underline{u}\in{\mathcal{S}}$. Moreover if all the
$u_{\alpha}$ are weak KAM solutions, then the inequality above turns out to be
an equality and $\underline{u}$ is a weak KAM solution.

In a similar manner,

$T^{-}\overline{u}(x)=\inf_{y\in X}\sup_{\alpha\in A}u_{\alpha}(y)+c(y,x)\\\
\geqslant\sup_{\alpha\in A}\inf_{y\in X}u_{\alpha}(y)+c(y,x)\\\
=\sup_{\alpha\in A}T^{-}u_{\alpha}(x)\leqslant\sup_{\alpha\in
A}u_{\alpha}(x)-{c[0]}=\overline{u}(x)-{c[0]}.$

∎

###  2.3 Regularity of subsolutions

In our discrete setting, the Aubry set enjoys the additional feature to be a
set where all subsolutions present systematic regularity properties. Away from
the Aubry set, this is false; subsolutions may fail to be continuous as was
shown in [174]:

######  Theorem 2.3.1.

Let $x\in X$ be a non–isolated point. Then $x\in{\mathcal{A}}$ if and only if
all subsolutions $u\in{\mathcal{S}}$ are continuous at $x$.

###### Proof.

If $x\in{\mathcal{A}}$ then we have seen that $T^{+}u-{c[0]}\leqslant
u\leqslant T^{-}u+{c[0]}$ and that equalities hold at $x$. As the left and
right terms of the inequalities are continuous, then the famous Sandwich
theorem implies that $u$ is continuous at $x$.

The hypothesis that $x$ is not isolated is used in the reciprocal (were $x$
isolated, $u$ could not be discontinuous at $x$). Assume now
$x\notin{\mathcal{A}}$. Let $u_{0}$ be the continuous subsolution constructed
in Theorem 1.4.1. In particular, the inequality $u\leqslant T^{-}u+{c[0]}$ is
now strict at $x$ and both functions are continuous. It follows from the
in–between lemma (proved after) that any function $v$ such that
$u_{0}\leqslant v\leqslant T^{-}u_{0}+{c[0]}$ will be a subsolution. In
particular, it can be constructed discontinuous at $x$, for example, take
$u_{0}=v$ everywhere, except at $x$ and $v(x)=T^{-}u_{0}(x)+{c[0]}$. ∎

######  Lemma 2.3.2.

Let $u$ be a subsolution and $v:X\to\mathbb{R}$ such that $u\leqslant
v\leqslant T^{-}u+{c[0]}$, then $v\in{\mathcal{S}}$.

###### Proof.

It follows from the monotonicity of the Lax–Oleinik semigroup:

$u\leqslant v\leqslant T^{-}u+{c[0]}\leqslant T^{-}v+{c[0]}.$

∎

###  2.4 More regularity for subsolutions

Of course, if more precise regularity results on subsolutions are aimed for,
some structure has to be added. Until further notice, we will take as base
space a compact smooth manifold $M$ and $c$ will be a cost on $M\times M$. Let
us recall a fundamental definition first:

######  Definition 2.4.1.

  1. (i)

Let $\Omega\subset\mathbb{R}^{n}$ be a convex open set. A function
$f:\Omega\to\mathbb{R}$ is said to be $K$–semiconcave if the function
$x\mapsto f(x)-K\|x\|^{2}$ is concave (the norm used is the Euclidean one). A
function is semiconcave111The notion we refer to here is sometimes called
semiconcave with linear modulus. For more details and proofs that are omitted
here see [60]. Other good references are also the unavoidable [93] and the
appendix of [94]. if it is $K$–semiconcave for some $K\in\mathbb{R}$.

  2. (ii)

A function $f:\Omega\to\mathbb{R}$ is locally semiconcave if each $x\in\Omega$
belongs to a neighborhood $V_{x}$ such that the restriction $f_{|V_{x}}$ is
semiconcave.

  3. (iii)

A function $f:M\to\mathbb{R}$ is locally semiconcave if for all coordinate
patch $\varphi:U\subset\mathbb{R}^{n}\to M$, the function $f\circ\varphi$ is
locally semiconcave ($n$ is assumed to be the dimension of $M$).

######  Remark 2.4.2.

The property of being locally semiconcave is invariant by $C^{2}$
diffeomorphisms. Therefore, in the previous definition, it is enough that
$f\circ\varphi_{i}$ be locally semiconcave for $\varphi_{i}:U_{i}\to M$ where
the $\varphi_{i}(U_{i})$ form any finite open cover of $M$. Of course, this
property is much easier to establish.

As concave functions can be characterized as functions whose graphs admit a
hyperplane tangent from above at every point, the following is implied:

######  Proposition 2.4.3.

A function $f:\Omega\to\mathbb{R}$ is $K$–semiconcave if and only if for all
$x\in\Omega$, there exists a linear form $p_{x}\in\mathbb{R}^{n*}$ such that

$\forall y\in\Omega,\quad f(y)\leqslant f(x)+p_{x}(y-x)+K\|y-x\|^{2}.$ (2.4)

The same holds true for locally semiconcave functions, but restricting to a
neighborhood of $x$ only. For locally semiconcave functions on a manifold $M$,
$p_{x}\in T_{x}^{*}M$ becomes an element of the cotangent fiber at $x$ and the
inequality is true in a chart.

We call superdifferential of a function $f$, assumed to be semiconcave (resp.
locally semiconcave), at $x$ (denoted $\partial^{+}f(x)$) the set of $p_{x}$
such that (2.4) holds (resp. in a neighborhood of $x$ or in a chart).

We state without proof:

######  Proposition 2.4.4.

Let $f:M\to\mathbb{R}$ be locally semiconcave and $x\in\mathbb{R}$. Then
$\partial^{+}f(x)$ is not empty, closed and convex. Moreover, $f$ is
differentiable at $x$ if and only if $\partial^{+}f(x)$ is a singleton (which
then contains only $D_{x}f$).

A very easy, though important, property of semiconcave functions is:

######  Proposition 2.4.5.

If $(f_{\alpha})_{\alpha\in A}$ is a family of $K$–semiconcave functions on
$U\subset\mathbb{R}^{n}$ then $\inf\limits_{\alpha\in A}f_{\alpha}$ is
$K$–semiconcave as soon as it is well defined.

The proof follows bearly the analogous property of concave functions. Of
course, each notion defined previously has a semiconvex counterpart which is
defined by replacing concave by convex and $-$ signs by $+$. The opposite of a
semiconcave function is then semiconvex and vice versa. A semiconvex function
$f$ has a subdifferential at each point denoted by $\partial^{-}f(x)$. It
coincides with $-\partial^{+}(-f)(x)$.

Motivated by the above, one defines sub–and superdifferentials for general
functions:

######  Definition 2.4.6.

Let $f:U\to\mathbb{R}$ be a function defined on an open set of
$\mathbb{R}^{n}$ the superdifferential  $\partial^{+}f(x)$ (resp.
subdifferential  $\partial^{-}f(x)$) of $f$ at $x\in U$ is the set of
$D_{x}\varphi$ where $\varphi:U\to\mathbb{R}$ is differentiable at $x$ and
verifies that $\varphi\geqslant f$ (resp. $\varphi\leqslant f$) with equality
at $x$.

######  Remark 2.4.7.

In the previous definition, the functions $\varphi$ can be taken equivalently
$C^{1}$. Sub–and superdifferentials are convex and closed. Moreover, $f$ is
differentiable at $x$ if and only if they are both non–empty. In this case,
$\partial^{+}f(x)=\partial^{-}f(x)=\\{D_{x}f\\}$. As a locally semiconcave
function has non–empty superdifferentials, we infer that if $f$ is locally
semiconcave, then $f$ is differentiable at $x$ if and only if
$\partial^{-}f(x)$ is non–empty.

Here is a not so obvious property that explains the nature of some results:

######  Proposition 2.4.8.

A function $f:M\to\mathbb{R}$ is $C^{1,1}$ (differentiable with Lipschitz
differential) if and only if it is both locally semiconcave and locally
semiconvex.

######  Definition 2.4.9.

A family of functions $f_{\alpha}:M\to\mathbb{R}$ for $\alpha\in A$ is said
equi–locally semiconcave if $M$ can be covered by finitely many open charts
$\varphi_{i}(U_{i})$, where $U_{i}\subset\mathbb{R}^{n}$ and if there are
constants $K_{i}$ such that all $f_{\alpha}\circ\varphi_{i}$ are
$K_{i}$–semiconcave.

Hypothesis: In the rest of this section, we assume that the families
$c(x,\cdot)$ and $c(\cdot,x)$ for $x\in M$ are equi–locally semiconcave.

It can be checked (as $M$ is compact) that a particular case of this is when
the function $c$ is itself locally semiconcave on $M\times M$.

When they exist, $\partial_{1}c(x,y)$ and $\partial_{2}c(x,y)$ denote the
partial derivatives of $c$ at $(x,y)$ with respect to the first and second
variable.

Coming back to discrete weak KAM theory, a refined version of Proposition
1.1.3 (i) then becomes:

######  Proposition 2.4.10.

Under the previous hypotheses, the image of $T^{-}$ (resp. $T^{+}$) consists
of equi–locally semiconcave (resp. equi–locally semiconvex) functions.

The proof is nothing but a direct application of Proposition 2.4.5. Note that
the result for $T^{+}$ involves semiconvexity because of the minus sign in its
definition.

As consequences, let us derive some further regularizing properties of the
Lax–Oleinik semigroups:

######  Proposition 2.4.11.

Let $v:M\to\mathbb{R}$ be a continuous function and $x_{0}\in M$. Let
$y_{0}\in M$ verify that $T^{-}v(x_{0})=v(y_{0})+c(y_{0},x_{0})$ (resp.
$T^{+}v(x_{0})=v(y_{0})-c(x_{0},y_{0})$). Then

  * •

$\partial^{+}_{2}c(y_{0},x_{0})\subset\partial^{+}T^{-}v(x_{0})$ (resp.
$-\partial^{+}_{1}c(x_{0},y_{0})\subset\partial^{-}T^{+}v(x_{0})$)222By
$\partial^{+}_{2}c(y_{0},x_{0})$ we mean the superdifferential of the map
$x\mapsto c(y_{0},x)$ at $x_{0}$..

  * •

In particular, if $T^{-}v$ (resp. $T^{+}v$) is differentiable at $x_{0}$ then
$\partial_{2}c(y_{0},x_{0})$ (resp. $-\partial^{+}_{1}c(x_{0},y_{0})$) exists
and $D_{x_{0}}T^{-}v=\partial_{2}c(y_{0},x_{0})$ (resp.
$-\partial^{+}_{1}c(x_{0},y_{0})\subset D_{x_{0}}T^{+}v$).

  * •

If $v$ is locally semiconcave (resp. semiconvex) then
$D_{y_{0}}v=-\partial_{1}c(y_{0},x_{0})$ (resp.
$D_{y_{0}}v=\partial_{2}c(x_{0},y_{0})$) and all the previous quantities do
exist.

###### Proof.

We prove half of the results leaving the rest as an exercise.

The first point is a direct consequence of the inequality

$\forall x\in M,\quad T^{-}v(x)\leqslant v(y_{0})+c(y_{0},x),$

which is an equality for $x=x_{0}$.

The second point then stems from the proved inclusion
$\partial^{+}_{2}c(y_{0},x_{0})\subset\partial^{+}T^{-}v(x_{0})$. By
hypothesis, the right hand side is a singleton and the left hand side is not
empty, hence they coincide and we get the result.

For the last part, note that the function $\varphi:y\mapsto v(y)+c(y,x_{0})$
reaches its minimum at $y_{0}$. Hence $0\in\partial^{-}\varphi(y_{0})$. But by
hypothesis, $\varphi$ is locally semiconcave and it is easily verified that
$\partial^{+}v(y_{0})+\partial^{+}_{1}c(y_{0},x_{0})\subset\partial^{+}\varphi(y_{0})$.
We infer that $\varphi$ is differentiable at $y_{0}$ with $D_{y_{0}}\varphi=0$
and that necessarily, $\partial^{+}v(y_{0})$ and
$\partial^{+}_{1}c(y_{0},x_{0})$ are singletons. Hence the result. ∎

######  Remark 2.4.12.

The following results were actually proven and used: if $f:M\to\mathbb{R}$ and
$g:M\to\mathbb{R}$ are locally semiconcave functions then

  * •

$f+g$ is differentiable at some $x\in M$ if and only if both $f$ and $g$ are;

  * •

if $f+g$ reaches a local minimum at some $x\in M$, then $f$ and $g$ are
differentiable at $x$.

We have now the necessary material to state a more precise version of Theorem
2.3.1:

######  Theorem 2.4.13.

Let $x\in{\mathcal{A}}$. Then any subsolution $u$ is differentiable at $x$.
Moreover, $D_{x}u$ does not depend on $u$.

###### Proof.

We will use the same inequalities as in Theorem 2.3.1. Let
$u\in{\mathcal{S}}$, then $T^{+}u-{c[0]}\leqslant u\leqslant T^{-}u+{c[0]}$.
Moreover those inequalities are equalities at $x\in{\mathcal{A}}$. This proves
that both $\partial^{-}u(x)\neq\varnothing$ (as $T^{+}u$ is locally
semiconvex) and $\partial^{+}u(x)\neq\varnothing$ (as $T^{-}u$ is locally
semiconcave). Hence $u$ is differentiable at $x$. Note that $T^{-}u$ and
$T^{+}u$ being subsolutions they are also differentiable at $x$ and the first
inequality above implies that all differentials are equal at $x$:
$D_{x}u=D_{x}T^{-}u=D_{x}T^{+}u$.

It remains to compute this differential. Let
$(x^{\prime},x)\in\widehat{\mathcal{A}}$. We know that $u(z)\leqslant
u(x^{\prime})+c(x^{\prime},z)+{c[0]}$ for all $z\in M$ and equality holds at
$z=x$. As $u$ is differentiable at $x$, this implies that the function
$u(x^{\prime})+c(x^{\prime},\cdot)+{c[0]}$ has a subdifferential at $x$, as it
is locally semiconcave, it is differentiable and its differential is
$\partial_{2}c(x^{\prime},x)$. Hence we conclude that
$D_{x}u=\partial_{2}c(x^{\prime},x)$ which happens to be independent on
$u\in{\mathcal{S}}$. ∎

######  Remark 2.4.14.

A similar proof implies that if $(x,y)\in\widehat{\mathcal{A}}$ and
$u\in{\mathcal{S}}$ then $D_{x}u=-\partial_{1}c(x,y)$. But this is not
surprising, as $x$ verifies $c(x^{\prime},x)+c(x,y)=\min\limits_{z\in
M}c(x^{\prime},z)+c(z,y)$.

Let us stress one more time that $c$ admits partial derivatives on the
$2$–Aubry set, as was actually established.

We now turn to improving Theorem 1.4.1:

######  Theorem 2.4.15.

There exists a strict subsolution $u_{1}$ which is $C^{1,1}$.

The proof makes crucial use of Ilmanen’s lemma (see [114, 102, 36, 46]).

######  Theorem 2.4.16 (Ilmanen’s lemma).

Given two functions $f,g:M\to\mathbb{R}$ such that $f$ is locally semiconvex,
$g$ is locally semiconcave and $f\leqslant g$, there exists a function $h$
which is $C^{1,1}$ such that $f\leqslant h\leqslant g$.

Moreover, if $h_{0}$ is a continuous function such that $f\leqslant
h_{0}\leqslant g$, then $h$ can be constructed arbitrarily close from $h_{0}$.

###### Proof of Theorem 2.4.15.

The proof splits into two steps. First we construct $C^{1,1}$ subsolutions,
then we explain how to make them strict.

Let us start with a subsolutions $u$. Then we have seen that $g=T^{-}u$ is a
locally semiconcave subsolution, $f=T^{+}g-{c[0]}$ is a locally semiconvex
subsolution and $f\leqslant g$. By Ilmanen’s lemma, there exists a $C^{1,1}$