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%
\def\bar#1{\overline#1}

\def\Cof{{\rm Cof \;}}
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\newtheorem{remark}{\bf Remark} 
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\def\thecor {\thesection\arabic{cor}}
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\def\thedefi {\thesection\arabic{defi}}
\def\theprop {\thesection\arabic{prop}}
\def\theremark {\thesection\arabic{remark}}
\def\theexample {\thesection\arabic{example}}
\def\theprob {\thesection\arabic{prob}}



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\def\debproof { \noindent{\sc Proof.~}}
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\def\debprop#1#2finprop {\begin{prop}#1 #2 \augmentetout \addtocounter{prop}{-1}\end{prop} }


\def\debcor#1#2fincor {\begin{cor}#1 #2 \augmentetout \addtocounter{cor}{-1}\end{cor} }


\def\debdefcor#1#2findefcor {\begin{defcor}#1 #2 \augmentetout \addtocounter{defcor}{-1}\end{defcor} }


\def\debprob#1#2finprob{\begin{prob}#1 #2  \augmentetout \addtocounter{prob}{-1}\end{prob} }


\def\debdef#1#2findef {\begin{defi}#1 #2  \augmentetout \addtocounter{def}{-1}\end{defi} }


\def\debrem#1#2finrem{ \begin{remark}#1 #2 \augmentetout \addtocounter{rem}{-1}\end{remark}}


\def\debex{\begin{example} \rm\mbox{}\augmentetout \addtocounter{example}{-1}\\}
\def\finex{\end{example}}

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\def\finbox{\end{minipage}\\ \\ \hline \end{tabular}}


%%%%
\catcode`@=11
\def\thebibliography#1{\noindent {\bf References}\sloppy\nolinebreak
 \list
 {[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth
 \advance\leftmargin\labelsep
 \usecounter{enumi}}
 \def\newblock{\hskip .11em plus .33em minus -.07em}
 \sloppy
 \sfcode`\.=1000\relax}
\catcode`@=12
%%%%%%%%%%%%%%
%%%%%%%%%%%%%%
% headings
%%%%
\catcode`@=11
\def\chaptermark#1{\markboth {\itpetit \ifnum \c@secnumdepth >\m@ne
 \@chapapp\ \thechapter. \ \fi #1}{}}
\def\sectionmark#1{\markright {\itpetit \ifnum \c@secnumdepth >\z@
 \thesection \ \ \fi  #1  }}
\catcode`@=12
%%%%%%%%%%%%%%
% figures
%%%%
%\catcode`@=11
%\def\fnum@figure{{\bf Figure \thefigure\ }}
%\catcode`@=12
%

%%Definitions utiles pour inserer des dessins en postscript, aldjia%%
\def\epsfx#1#2{\leavevmode\epsfxsize=#1 \epsfbox{#2}}
\def\epsfy#1#2{\leavevmode\epsfysize=#1 \epsfbox{#2}}
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%$$ \hbox{\hss\small Figure \the\clegende~: #1 \hss} $$}
%$$ \hbox{\hss\small #1 \hss} $$}       si on ne veut pas les figures numerotees 

%\newcommand{\epslabnew}[3]
%           { \vspace{0.5cm} 
%                           \begin{tabular}{c}
%                           {\epsfx{#2}{#1} } \\
%             \vspace{0.3cm}
%                           {\legende{\em{#3}}} \\
%                                \end{tabular}
%                                \vspace{0.2cm} 
%                           }  

\def\clegende#1#2{
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%POUR FAIRE DES FIGURES

%tout ce qui suit est extrait de Fichier epsf.tex:

%   EPSF.TEX macro file:
%   Written by Tomas Rokicki of Radical Eye Software, 29 Mar 1989.
%   Revised by Don Knuth, 3 Jan 1990.
%   Revised by Tomas Rokicki to accept bounding boxes with no
%      space after the colon, 18 Jul 1990.
%
%   TeX macros to include an Encapsulated PostScript graphic.
%   Works by finding the bounding box comment,
%   calculating the correct scale values, and inserting a vbox
%   of the appropriate size at the current position in the TeX document.
%
%   To use with the center environment of LaTeX, preface the \epsffile
%   call with a \leavevmode.  (LaTeX should probably supply this itself
%   for the center environment.)
%
%   To use, simply say
%   \input epsf           % somewhere early on in your TeX file
%   \epsfbox{filename.ps} % where you want to insert a vbox for a figure
%
%   Alternatively, you can type
%
%   \epsfbox[0 0 30 50]{filename.ps} % to supply your own BB
%
%   which will not read in the file, and will instead use the bounding
%   box you specify.
%
%   The effect will be to typeset the figure as a TeX box, at the
%   point of your \epsfbox command. By default, the graphic will have its
%   `natural' width (namely the width of its bounding box, as described
%   in filename.ps). The TeX box will have depth zero.
%
%   You can enlarge or reduce the figure by saying
%     \epsfxsize=<dimen> \epsfbox{filename.ps}
%   (or
%     \epsfysize=<dimen> \epsfbox{filename.ps})
%   instead. Then the width of the TeX box will be \epsfxsize and its
%   height will be scaled proportionately (or the height will be
%   \epsfysize and its width will be scaled proportiontally).  The
%   width (and height) is restored to zero after each use.
%
%   A more general facility for sizing is available by defining the
%   \epsfsize macro.    Normally you can redefine this macro
%   to do almost anything.  The first parameter is the natural x size of
%   the PostScript graphic, the second parameter is the natural y size
%   of the PostScript graphic.  It must return the xsize to use, or 0 if
%   natural scaling is to be used.  Common uses include:
%
%      \epsfxsize  % just leave the old value alone
%      0pt         % use the natural sizes
%      #1          % use the natural sizes
%      \hsize      % scale to full width
%      0.5#1       % scale to 50% of natural size
%      \ifnum#1>\hsize\hsize\else#1\fi  % smaller of natural, hsize
%
%   If you want TeX to report the size of the figure (as a message
%   on your terminal when it processes each figure), say `\epsfverbosetrue'.
%
\newread\epsffilein    % file to \read
\newif\ifepsffileok    % continue looking for the bounding box?
\newif\ifepsfbbfound   % success?
\newif\ifepsfverbose   % report what you're making?
\newdimen\epsfxsize    % horizontal size after scaling
\newdimen\epsfysize    % vertical size after scaling
\newdimen\epsftsize    % horizontal size before scaling
\newdimen\epsfrsize    % vertical size before scaling
\newdimen\epsftmp      % register for arithmetic manipulation
\newdimen\pspoints     % conversion factor
%
\pspoints=1bp          % Adobe points are `big'
\epsfxsize=0pt         % Default value, means `use natural size'
\epsfysize=0pt         % ditto
%
\def\epsfbox#1{\global\def\epsfllx{72}\global\def\epsflly{72}%
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   \def\lbracket{[}\def\testit{#1}\ifx\testit\lbracket
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%
\def\epsfgetlitbb#1#2 #3 #4 #5]#6{\epsfgrab #2 #3 #4 #5 .\\%
   \epsfsetgraph{#6}}%
%
\def\epsfnormal#1{\epsfgetbb{#1}\epsfsetgraph{#1}}%
%
\def\epsfgetbb#1{%
%
%   The first thing we need to do is to open the
%   PostScript file, if possible.
%
\openin\epsffilein=#1
\ifeof\epsffilein\errmessage{I couldn't open #1, will ignore it}\else
%
%   Okay, we got it. Now we'll scan lines until we find one that doesn't
%   start with %. We're looking for the bounding box comment.
%
   {\epsffileoktrue \chardef\other=12
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%
%   We check to see if the first character is a % sign;
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%   if so, we look further and stop only if the line begins with
%   `%%BoundingBox:'.
%
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%
%   Now we have to calculate the scale and offset values to use.
%   First we compute the natural sizes.
%
\def\epsfclipstring{}% do we clip or not?  If so,
\def\epsfclipon{\def\epsfclipstring{ clip}}%
\def\epsfclipoff{\def\epsfclipstring{}}%
%
\def\epsfsetgraph#1{%
   \epsfrsize=\epsfury\pspoints
   \advance\epsfrsize by-\epsflly\pspoints
   \epsftsize=\epsfurx\pspoints
   \advance\epsftsize by-\epsfllx\pspoints
%
%   If `epsfxsize' is 0, we default to the natural size of the picture.
%   Otherwise we scale the graph to be \epsfxsize wide.
%
   \epsfxsize\epsfsize\epsftsize\epsfrsize
   \ifnum\epsfxsize=0 \ifnum\epsfysize=0
      \epsfxsize=\epsftsize \epsfysize=\epsfrsize
      \epsfrsize=0pt
%
%   We have a sticky problem here:  TeX doesn't do floating point arithmetic!
%   Our goal is to compute y = rx/t. The following loop does this reasonably
%   fast, with an error of at most about 16 sp (about 1/4000 pt).
% 
     \else\epsftmp=\epsftsize \divide\epsftmp\epsfrsize
       \epsfxsize=\epsfysize \multiply\epsfxsize\epsftmp
       \multiply\epsftmp\epsfrsize \advance\epsftsize-\epsftmp
       \epsftmp=\epsfysize
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          \ifnum\epsftsize<\epsfrsize\else
             \advance\epsftsize-\epsfrsize \advance\epsfxsize\epsftmp \fi
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       \epsfrsize=0pt
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     \epsftmp=\epsfrsize \divide\epsftmp\epsftsize
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     \multiply\epsftmp\epsftsize \advance\epsfrsize-\epsftmp
     \epsftmp=\epsfxsize
     \loop \advance\epsfrsize\epsfrsize \divide\epsftmp 2
     \ifnum\epsftmp>0
        \ifnum\epsfrsize<\epsftsize\else
           \advance\epsfrsize-\epsftsize \advance\epsfysize\epsftmp \fi
     \repeat
     \epsfrsize=0pt
    \else
     \epsfrsize=\epsfysize
    \fi
   \fi
%
%  Finally, we make the vbox and stick in a \special that dvips can parse.
%
   \ifepsfverbose\message{#1: width=\the\epsfxsize, height=\the\epsfysize}\fi
   \epsftmp=10\epsfxsize \divide\epsftmp\pspoints
   \vbox to\epsfysize{\vfil\hbox to\epsfxsize{%
      \ifnum\epsfrsize=0\relax
        \special{PSfile=#1 llx=\epsfllx\space lly=\epsflly\space
            urx=\epsfurx\space ury=\epsfury\space rwi=\number\epsftmp
            \epsfclipstring}%
      \else
        \epsfrsize=10\epsfysize \divide\epsfrsize\pspoints
        \special{PSfile=#1 llx=\epsfllx\space lly=\epsflly\space
            urx=\epsfurx\space ury=\epsfury\space rwi=\number\epsftmp\space
            rhi=\number\epsfrsize \epsfclipstring}%
      \fi
      \hfil}}%
\global\epsfxsize=0pt\global\epsfysize=0pt}%
%
%   We still need to define the tricky \epsfaux macro. This requires
%   a couple of magic constants for comparison purposes.
%
{\catcode`\%=12 \global\let\epsfpercent=%\global\def\epsfbblit{%BoundingBox}}%
%
%   So we're ready to check for `%BoundingBox:' and to grab the
%   values if they are found.
%
\long\def\epsfaux#1#2:#3\\{\ifx#1\epsfpercent
   \def\testit{#2}\ifx\testit\epsfbblit
      \epsfgrab #3 . . . \\%
      \epsffileokfalse
      \global\epsfbbfoundtrue
   \fi\else\ifx#1\par\else\epsffileokfalse\fi\fi}%
%
%   Here we grab the values and stuff them in the appropriate definitions.
%
\def\epsfempty{}%
\def\epsfgrab #1 #2 #3 #4 #5\\{%
\global\def\epsfllx{#1}\ifx\epsfllx\epsfempty
      \epsfgrab #2 #3 #4 #5 .\\\else
   \global\def\epsflly{#2}%
   \global\def\epsfurx{#3}\global\def\epsfury{#4}\fi}%
%
%   We default the epsfsize macro.
%
\def\epsfsize#1#2{\epsfxsize}
%
%   Finally, another definition for compatibility with older macros.
%
\let\epsffile=\epsfbox



\def\T{\mathbf{T}}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\Arctg{\mathrm{Arctg\ }}
\def\di{\displaystyle}
\def\bs{\bigskip}
\def\cotg{\mathrm{cotg\ }}
\def\e{\varepsilon}
\def\nhabla{\hat\nabla}
\def\Utilde{\tilde{U}}
\def\utilde{\tilde{u}}
\def\R2+{\RR ^2_+}
\def\ms{\medskip}
\def\pref#1{{\rm \ref{#1}}}
\def\fref#1{{\rm (\ref{#1})}}
\def\Log{\mbox{Log}}
\def\dundemi{\partial_{1/2}}
\def\Iundemi{I_{1/2}}
\def\ni{\noindent}
\def\Htd{H^{3/2}}
\def\Hud{H^{1/2}}
\def\Hcd{H^{5/2}}
\def\ou{\overline u}
\def\ue{u^\varepsilon}
\def\uu{\underline u}
\def\pe{p^\varepsilon}
\def\Uepsn{U^\varepsilon_{(n)}}
\def\uepsn{u^\varepsilon_{(n)}}
\def\veps{v^\varepsilon}
\def\Veps{V^\varepsilon}
\def\Ueps{U^\varepsilon}
\def\beps{b^\varepsilon}
\def\Beps{B^\varepsilon}
\def\psieps{\psi^\varepsilon}
\def\psiepsn{\psi^\varepsilon_{(n)}}
\def\Lambdaeps{\Lambda^\varepsilon}
\def\aeps{a^\varepsilon}
\def\deltaeps{\delta^\varepsilon}
\def\Phibar{\overline{\Phi}}
\def\Czeroalpha{C^{0,\alpha}}
\def\Cunalpha{C^{1,\alpha}}
\def\Cdeuxalpha{C^{2,\alpha}}
\def\LL{{\cal L}}
\def\M{{\cal M}}
\def\D{{\cal D}}
\def\Ctroisalpha{C^{3,\alpha}}
\def\Cinfini{C^{\infty}}
\def\Linfini{L^{\infty}}
\def\Wuninfini{W^{1,\infty}}
\def\Wunquatre{W^{1,4}}
\def\Wunun{W^{1,1}}
\def\NN{I\!\!N}
\def\RR{I\!\!R}

\def\intRR{\int_{\RR}}
\def\intR2+{\int_{\R2+}}
\def\phiz{\phi_0}
\def\phiu{\phi_1}
\def\Du{{{\cal D}_1}}
\def\Dd{{{\cal D}_2}}
\def\Dt{{{\cal D}_3}}
\def\TT{{\cal T}}
\def\TTT{{\mathbf T}}
\def\th{\mathrm{th}}
\def\S{{\cal S}}
\def\tS{\tilde{\cal S}}
\def\lto{\ensuremath{\longrightarrow}}
\def\A{{\cal A}}

\def\tg{\mathrm{tg}\ }
\def\grandS{\mbox{\large $ \cal S $}}

\def\Hb{\overline H}
\def\Gamb{\overline \Gamma}
\def\Z{Z\!\!\!Z}
\def\reff#1{(\ref{#1})}
\def\ub{\overline u}
\def\chib{\overline \chi}
\def\1{1\!\!1}

\def\cper{{\cal C}_{per}}
\def\xoe{{\displaystyle \frac{x}{\varepsilon}}}
\def\yoe{{\displaystyle \frac{y}{\varepsilon}}}
\def\toe{{\displaystyle \frac{t}{\varepsilon}}}

%%% fin des alias
\begin{document}
\begin{center}

\centerline{\Large \bf Convergence to time-periodic solutions in}
\centerline{\Large \bf Hamilton-Jacobi equations on the circle}


\bigskip

{\bf Patrick BERNARD}

\medskip {\footnotesize 
Institut Fourier, UMR CNRS 5582
Universit\'e de Grenoble I 
BP 74, 38402 Saint-Martin d'H\`eres, France 
}


\bigskip

{\bf Jean-Michel ROQUEJOFFRE}

\medskip {\footnotesize UFR-MIG, UMR CNRS 5640.  Universit\'e Paul
Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex, France}

\rule{30mm}{.1mm}
\end{center}

\ni {\small {\sc Abstract.} \rule [.5mm]{2mm}{.1mm} 
.} {\small The goal of this paper is to give a simple  proof of the convergence to
time-periodic states  of the solutions of time-periodic
Hamilton-Jacobi equations on the circle with convex Hamiltonian.
Note that the period of limiting solutions may be greater than the period 
of the Hamiltonian.}


\ni {\small {\sc Resum\'e.} \rule [.5mm]{2mm}{.1mm} 
.} {\small On donne une preuve simple de la convergence vers des \'etats
p\'eriodiques en temps pour les solutions d'\'equations de Hamilton-Jacobi
sur le cercle avec un  Hamiltonien convexe et p\'eriodique en temps.
Il est \`a noter que la p\'eriodes des solutions limites
peut \^etre plus grande que la p\'eriode du Hamiltonien.}



\Section{Introduction}
We consider the Hamilton-Jacobi equation
\beq
\label{HJ}
u_t+H(t,x,u_x)=0,\ \ \ x\in \T
\eeq
where $\T$ is the unit circle. 
The Hamiltonian $H(t,x,p):\ \RR\times\T\times\RR\mapsto\RR$
is $C^2$, 1-periodic in $t$, 
and satisfies the following classical hypotheses:
\begin{itemize}
\item
Strict convexity: $H_{pp}(t,x,p)>0$ for all 
$(t,x,p)\in\RR\times\T\times\RR.$
\item
Super-linearity: $H(t,x,p)/p\lto \infty$ as $|p|\lto \infty$
for each $(t,x)\in \RR\times\T$.
\item
Completeness: The Hamiltonian vector-field
$$X(t,x,p)=(H_p(t,x,p),-H_x(t,x,p))$$
is complete, \textit{i.e.}
for all $(t_0,x_0,p_0)$, there exists a $C^2$ curve
$\gamma(t)=(x(t),p(t)):\RR\lto \T\times \RR$
such that $(x(t_0),p(t_0))=(x_0,p_0)$
and $\dot \gamma(t)=X(t,\gamma(t))$ for all $t\in\RR$.
\end{itemize}
The first two assumptions are classical in the viscosity solutions theory;
see \cite{PL2}. The last one is introduced  in Mather \cite{Mather}; note that it is satisfied if
there exists a constant $C$ such that
$
|H_t|\leq C(1+H)$.

Under the above three assumptions, the Cauchy Problem for (\ref{HJ}) is well posed in the viscosity sense:
given a time $s\in \RR$ and a continuous function $u_0:\ \T\to\RR$, equation
(\ref{HJ}) has a unique viscosity solution 
$u(t,x):[s,+\infty[\times\T\lto \RR$, such that $u(s,.)=u_0$.
It will be denoted by
$\TT(s,t)u_0$. See \cite{PL2}, for instance.

It is known - and this is not specific to the one-dimensional setting - 
that 
there exists a real  number $\lambda$
such that 
$u(t,x)+\lambda t$
is bounded for all viscosity solution $u:[s,+\infty[\times\T\lto \RR$
of (\ref{HJ}).
The real number $\lambda$ has various  different names:
It is the critical value of Ma\~ne,
see \cite{Mane}, \cite{CDI} or the value 
$\alpha(0)$, where $\alpha:\RR\lto\RR$ is the 
Mather function, see \cite{Mather}, or the averaged Hamiltonian \cite{LPV}.
Note that the number $\lambda$ may also be viewed
as the eigenvalue of the Hopf-Lax-Oleinik operator
in the sense of min-plus algebra, see \cite{S}.
We are  interested in proving the following result.




\begin{thm}
\label{t1.1}
Let $u(t,x):[s,+\infty[\times\T\lto \RR$ be a viscosity solution
of (\ref{HJ}).
There exist an integer $T$ and a viscosity solution 
$l(t,x)=\lambda t + \phi(t,x):\RR\times\T\lto\RR$
such that $\phi$ is $T$-periodic in $t$ and
$$
\lim _{t\lto \infty }\big\|u(t,.)-l(t,.)\big\|_{\infty}= 0.
$$
\end{thm}

In the following, we will always assume
that $\lambda=0$, which can be obtained by replacing
the Hamiltonian $H$
by $H-\lambda$.

It is known that there always exist a viscosity solution
of (\ref{HJ}) which is $1$-periodic in time.
However, it is not hard to build examples of
viscosity solutions of equations of the form 
(\ref{HJ}) which do not converge to $1$-periodic
solutions, see  \cite{BS2} and \cite{FaMa}.
More precisely, one can build solutions
which are periodic in time, but 
of minimal period greater than one.
Hence one cannot expect to have 
always $T=1$ in the theorem.

For time-independent Hamiltonians, convergence to steady states is known: a particular nontrivial multidimensional case is studied in 
\cite{NR2}, and the general result in one dimension 
is given in \cite{R2}. The general multidimensional result is due to Fathi \cite{Fa3}, and two different proofs are available in 
\cite{R} and \cite{BS2}.

The situation is not so clear when the Hamiltonian is time-periodic.
In order to be more precise, it is useful to
recall that one can associate to the equation
(\ref{HJ}) a rotation number $\rho\in \RR$, see section \ref{circle},
  which is the 
rotation number of extremals.
We then have the following





\textbf{Addendum}
\begin{itshape}
The period $T$ 
in the theorem is $1$ if the rotation number $\rho$
is irrational and is not greater  than $q$ if 
$\rho$ is a rational $p/q$.
\end{itshape}



The theorem and its addendum have been proved in 
 \cite{Be} from the dynamical system point of view,
that is from the study of extremals.



The result in the case of a rational rotation number
had been previously obtained by a method
relying on the dynamic programming principle
in  \cite{R}.
It turns out that the ideas of this paper
can be exploited further to provide
a simpler proof of the general case.
It is our aim to present this proof here.

We shall first recall the general properties
of viscosity solutions, in Section \ref{general}.
In Section \ref{calibrated}, we introduce
some dynamics, define some Aubry-Mather sets,
and recall 
specific observations concerning $\omega$-limit solutions,
mostly taken from \cite{R}.
All the results in these sections are general, 
and remain true if one considers equation
(\ref{HJ}) on any compact manifold.
We complement these general observations
by specific one-dimensional arguments
in Section \ref{circle} to conclude the proof.

%
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%
%
\Section{General properties and a large time behaviour candidate}
\label{general}
%
%
%
%
%
%
%
Let us start with some well understood properties of viscosity solutions without proof, see \cite{PL2}, \cite{Fa1}
or \cite{FaMa}.
It is useful to introduce the Lagrangian
$$
L(t,x,v)=\max_{p\in\RR} \biggl(pv-H(t,x,p)\biggl),
$$
which is also  convex and superlinear.
We have the following fundamental result of the calculus of variations
(see a proof in the appendix of \cite{Mather}):

Let us fix a time interval $[t,t']$, and two points
$X,X'$ in $\T$.
If the Lagrangian is convex and superlinear,
the action integral
$$
\int _t ^{t'} L(s,x(s),\dot x(s)) ds
$$
reaches its minimum on the set of 
absolutely continuous curves $x(s):[t,t']\lto  \T$
which satisfy $x(t)=X$ and $x(t')=X'$.
If in addition the Hamiltonian flow  is complete, then the minimum
is reached by $C^2$ curves which  satisfy the Euler-Lagrange equations.
Denoting 
$
p(s)= L_v(q(s),\dot q(s),s),$
these equations can be written
$$
\dot p(s)= L_x(s,x(s),\dot x(s))=
 H_x(s,x(s),p(s))
$$
hence the curve $(x(s),p(s))$ is a trajectory of 
the Hamiltonian vector-field.







Let us define, for all $t>s$, the 
Hopf-Lax-Oleinik operator $\TT(s,t)$, which, to any continuous 
function $u_0:\T\lto \RR$, associates the continuous function


\beq
\label{LO}
\TT(s,t)u_0(x)=\inf_{\gamma}\biggl(u_0(\gamma(s))+\int_s^tL(\sigma,\gamma,\dot\gamma)\ d\sigma\biggl),
\eeq
%
%
%
where the infimum is taken on the set of absolutely continuous 
curves $\gamma:[s,t]\lto \T$ satisfying $\gamma(t)=x$. 


The viscosity solution  $u(t,x)\in C([s,+\infty[\times\T,\RR)$
of (\ref{HJ}) with initial condition 
 $u(s,.)=u_0$
is given by $u(t,.)=\TT(0,t)u_0$.
This may be taken as the definition of viscosity solutions
in the present paper. This definition 
is equivalent to the standard one in our setting,
see \cite{FaMad} for a clear proof of this
piece of folklore.



We have the Markov property
$$
 \TT(t,t')\circ\TT(s,t)= \TT(s,t')
$$
for $s\leq t\leq t'$,
hence the mappings $\TT(0,n)=\TT(0,1)^n$, $n\in\NN$
form a discrete semi-group.
We will note $\TT$ for $\TT(0,1)$ for simplicity.
The following simple properties are proved in \cite{Fabook}
in the autonomous case, but the extension to the nonautonomous
case does not present any difficulty.




The mappings $\TT(s,t)$ are contractions,
$$
\big\|
\TT(s,t)u-\TT(s,t)v
\big\|_{\infty}
\leq
\big\|u-v
\big\|_{\infty}.
$$

The mappings  $\TT(s,t)$ are compact.
More precisely, 
there exists a positive nonincreasing function
$K(\epsilon):]0,\infty[\lto]0,\infty[$
such that $\TT(s,t)u$
is $K(\epsilon)$-Lipschitz for all continuous function $u$ 
and all $t\geq s+\epsilon$.


The mappings  $\TT(s,t)$ are order-preserving and satisfy
$\TT(s,t)(c+u)=c+\TT(s,t)(u)$ for all real $c$.



The following proposition will be useful in the sequel.
Note that it  implies the existence of a fixed point of
the semi-group generated by $\TT$,
\textit{i.e.} the existence of a $1$-periodic viscosity solution
of (\ref{HJ}).

\begin{prop}
\label{liminf}
Let $u(t,x)\in C([s,\infty[\times \T,\RR)$
be a viscosity solution of (\ref{HJ}); recall that it is bounded on 
$[s,\infty[\times\T$.
Then the $1$-periodic function
$$
\phi(t,x)=\liminf_{n\to+\infty}u(t+n,x)
$$
is a viscosity solution of (\ref{HJ}).
\end{prop}

We learn from the referee that this proposition is a particular case
of a result of Barron and Jensen,  see \cite{BJ}.
Note also that it is a very general result 
from Barles and Perthame that 
$\phi$ is a viscosity subsolution, see  \cite{BP}.
We shall however provide a simple proof using
the Hopf-Lax-Oleinik expression  of solutions.


\noindent{\sc Proof.} 
We have to prove that $\TT(s,t)\phi(s,.)=\phi(t,.)$
for all $s\leq t$.
Let us first prove that $\phi$ is a subsolution, \textit{i. e. }
that  $\TT(s,t)\phi(s,.)\leq \phi(t,.)$.
This fact, which is a general feature of viscosity solutions of 
first or second order equations, can be easily seen on  (\ref{LO}).
In order to do so, we 
fix $(t,x)$ and consider 
 an increasing sequence $n_k$
of integers such that $u(t+n_k,x)\lto \phi(t,x)$.
There exists a sequence of curves $\gamma_k:[s,t]\lto\T$ such that
$$
u(t+n_k,x)=u(s+n_k,\gamma_k(0))+
\int_s^{t}L(\sigma,\gamma_k(\sigma+t),\dot\gamma_k(\sigma+t))\ d\sigma.
$$
The sequence $\gamma_k$ is compact for the $C^1$ topology,
and we will assume by possibly taking a subsequence in $n_k$
that it is convergent, and note $\gamma$ the limit.
Taking the $\liminf$ in the equality above gives 
$$
\phi(t,x)\geq \phi(s,\gamma(0))+
\int_s^{t}L(\sigma,\gamma(\sigma+t),\dot\gamma(\sigma+t))\ d\sigma
\geq \TT(s,t)\phi(s,.)(x).
$$
We have used that the functions $u(s+n_k,.)$ have a common Lipschitz 
constant to conclude that $\liminf u(s+n_k, \gamma_k(0))=
\liminf u(s+n_k, \gamma(0))\geq \phi(s,\gamma(0))$.


The reverse inequality is specific to 
Hamilton-Jacobi equations, and explicitely 
relies on (\ref{LO}). 
Note that for all curve $\gamma:[s,t]\lto\T$,
we have 
$$
u(t+n,x)\leq u(s+n,\gamma(0))+
\int_s^{t}L(\sigma,\gamma(\sigma+t),\dot\gamma(\sigma+t))\ d\sigma.
$$
Taking the $\liminf$, we obtain 
$$
\phi(t,x)\leq \phi(s,\gamma(0))+
\int_s^{t}L(\sigma,\gamma(\sigma+t),\dot\gamma(\sigma+t))\ d\sigma
$$
for each curve $\gamma$,
hence $\phi(t,.)\leq \TT(s,t)\phi(s,.)$,
which is the desired inequality.
 \finproof




The basic objects to understand in order to
study  the asymptotic behaviour of
solutions of  (\ref{HJ}) are the $\omega$-limit solutions.
Recall that a solution $u(t,x):\RR\times \T\lto \RR$
is called an $\omega$-limit
solution if there exists a solution 
$v:[s,\infty[\times \T\lto\RR$
and an increasing sequence $n_k$ of integers such that
$$u(t,x)=\lim_{k\lto \infty}v(t+n_k,x).$$
In other words, $\omega$-limit solutions are solutions
whose initial value $u(0,.)$ is an $\omega$-limit
of the semi-group $\TT$.
%Note that we cannot infer, without any additional arguments,
%that the above constructed solution $\phi$ is 
%an $\omega$-limit solution.


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%


\section{Calibrated curves and Uniqueness set}
\label{calibrated}
%
%
%
%
%
%
In this section, we give - without many proofs, for they are already stated in \cite{Fa1} or \cite{R} - 
some salient features of $\omega$-limit solutions that do not depend of the dimension of the ambient space.

\medskip
Let $u:[s,\infty[\times\T\lto\RR$
be a viscosity solution of (\ref{HJ}).
A curve $\gamma:[s,\infty[\supset[t,t']\lto \T$
is said calibrated by $u$ if
$$
u(t',\gamma(t'))=u(t,\gamma(t))+
\int_{t}^{t'}L(\sigma,\gamma,\dot\gamma)\ d\sigma.
$$ 
Note that if $\gamma(s)$ is a calibrated curve,
then the curve $(\gamma(s),p(s))$ is a trajectory
of the Hamiltonian flow, where 
$p(s)=\partial_v L(s,\gamma(s),\dot \gamma(s))$.
We have the following regularity result
(fathi,  \cite{Fa1}):
If $\gamma:[t,t']\lto \T$
is calibrated by $u$,
then $u_x$ exists at each point $(s,\gamma(s)),s\in [t,t'[$
and satisfies
$$
u_x(s,\gamma(s))=p(s)=L_v(s,\gamma(s),\dot\gamma(s))
\Longleftrightarrow
\dot\gamma(s)=H_p(s,\gamma(s),u_x(s,\gamma(s)).
$$

It was already proved in \cite{Fleming}
that this relation holds at differentiability points of $u$.
The regularity of $u$  on calibrated curves 
was noticed  in \cite{PR} for the equation
$\vert\nabla u\vert=f(x)$ on the sphere, 
$f$ nonnegative with nonempty zero set.
It was not, however, made as systematic as in Fathi's work.


\medskip
We now choose once and for all a 
 1-periodic solution  $\phi(t,x)$ of (\ref{HJ}).
%i.e. the solution constructed in Proposition \ref{liminf}.
First, let us note that a classical compactness argument gives the existence of
curves $\gamma:\RR\lto\T$
which are calibrated by $\phi$ on all compact interval.

It is a consequence of  Fathi's regularity result
that two such curves cannnot intersect.
Indeed, if $\gamma_1$ and  $\gamma_2:\RR\lto \T$
are calibrated by $\phi$, and if there exists a $t$ such that
$\gamma_1(t)=\gamma_2(t)$, 
then $u_x$ exists at the point $(t,\gamma_1(t))$,
and, setting 
$p_i(s)=\partial L(s,\gamma_i(s),\dot \gamma_i(s))$,
we have 
$$
p_1(t)=u_x(t,\gamma_1(t))=u_x(t,\gamma_2(t))=p_2(t).
$$
The curves $(\gamma_i(s),p_i(s))$ are then two trajectories
of the Hamiltonian flow which are equal at time $t$, 
so that they coincide for all times.


Let  $$ \D\subset \RR\times \T$$
be the union of the graphs of these orbits,
and $\D_0\subset \T$
be the set of points $\gamma(0)$, where 
$\gamma:\RR\lto \T$ is calibrated.
This is a nonempty compact set.
For each $t$, we define the mapping $S^t:\D_0\lto \T$
which associates to each $x\in \D_0$,
the value $\gamma(t)$, where $\gamma:\RR\lto \T$
is the unique calibrated curve satisfying $\gamma(0)=x$.
It is a  bi-Lipschitz homeomorphism onto its image.

Clearly, $S^1$ is a homeomorphism of $\D_0$.
Let us note $\M_0$ its $\omega$-limit.
This is the closure in $\T$ 
of the set of points $x\in \D_0$
which are the limit of a sequence 
$S^{n_k}(y)$ with $y\in \D_0$ and 
$n_k$ an increasing sequence of integers.
The set $\M_0$ is non-empty and compact.
We call $\M$ the union, in $\RR\times\T$,
of the graphs  of curves $S^t(x),x\in \M_0$.



The following remark, noticed in 
\cite{R}, is the key point to the convergence proof:
\begin{lem}
 \label{decroit}
Let  $u(t,x):[s,\infty[\times \T\lto \RR$
be a  viscosity solution of (\ref{HJ}), let $x\in \D_0$.
Then
the function
 $ t\mapsto u(t,S^t(x))-\phi(t,S^t(x))$
 is non-increasing.
\end{lem}
The important consequence below is also proved in \cite{R}:
\begin{cor}
\label{const}
Let $u(t,x)$ be an $\omega$-limit viscosity solution of  (\ref{HJ}),
and let $x\in \M_0$,
then 
the function  $ t\mapsto u(t,S^t(x))-\phi(t,S^t(x))$
is constant.
\end{cor}

It follows that the curve $S^t(x)$ is calibrated by 
$u$; hence $u$ and $\phi$ are differentiable on $\M$ 
by the above-stated regularity results of Fathi. 
%A Taylor expansion 
%of $p\mapsto H(t,x,p)$ around $D\phi$, at a point 
%$(t,x)\in\M$, gives the 
\begin{cor}
Let $u(t,x)$ be an $\omega$-limit viscosity solution of  (\ref{HJ}).
Then the functions $u(t,x)$ and $\phi(t,x)$
and differentiable on $\M$, and we have
$$
\partial_t(u-\phi)(t,x)=\partial_x(u-\phi)(t,x)=0
$$
for all $(t,x)\in \M$.
\end{cor}

The next step is to define a uniqueness set, i.e. a set such that two global solutions of
(\ref{HJ}) coinciding on this set coincide everywhere. We formulate the results in the general, non-autonomous, setting.

\begin{prop}
\label{unique}
Let  $u(t,x):\RR\times \T\lto\RR$
be a global and bounded viscosity solution of (\ref{HJ})
such that $u=\phi$ on $\M$,
then $u=\phi$.
\end{prop}
%
%
This proposition easily follows from the following lemma:

\begin{lem}
Let  $u(t,x):\RR\times \T\lto\RR$
be a global and bounded viscosity solution of (\ref{HJ})
and $\gamma(t):]-\infty,s]\lto \T$
be a curve calibrated by $u$.
Then there exists an increasing sequence $n_k$
of integers such that
$\gamma(-n_k)\lto x\in \M_0$.
\end{lem}
%
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%
\noindent{\sc Proof.}
In view of Lemma \ref{decroit},
the function 
$t\longmapsto u(t,\gamma(t))-\phi(t,\gamma(t))$
is non-increasing, and bounded.
Hence this function has a limit as $t\lto -\infty$.
Let us choose an increasing  sequence $n_k$ of integers
such that the curves 
$\gamma(t-n_k)$
are converging uniformly on compact sets
to a limit $\gamma_{\infty}:\RR\lto\T$.
The following calculations show that 
this curve is calibrated by $\phi$:
$$
\phi(t',\gamma_{\infty}(t'))-\phi(t,\gamma_{\infty}(t))
=
\lim\Big(
\phi(t',\gamma(t'-n_k))-\phi(t,\gamma(t-n_k))
\Big)
$$
$$
=\lim\Big(
\phi(t'-n_k,\gamma(t'-n_k))-\phi(t-n_k,\gamma(t-n_k))\Big)
$$
$$
=\lim\Big(
u(t'-n_k,\gamma(t'-n_k))-u(t-n_k,\gamma(t-n_k))
\Big)
$$
$$
=\lim \int_{t}^{t'}L(\sigma,\gamma(\sigma-n_k),
\dot\gamma(\sigma-n_k)) d\sigma
=\int_{t}^{t'}L(\sigma,\gamma_{\infty}(\sigma),
\dot\gamma_{\infty}(\sigma)) d\sigma.
$$
As a consequence, the curve $t\lto (t,\gamma_{\infty}(t)) $
is asymptotic to $\M$,
so that there exists an increasing sequences $m_k$
of integers such that
that $\gamma_{\infty}(m_k)\lto x\in \M_0$.
Possibly taking a subsequence of $n_k$, we obtain
that $\gamma(m_k-n_k)\lto x\in \M.$
\finproof



In the autonomous case, where the Hamiltonian $H$
does not depend on the variable $t$, the above remarks 
imply that any $\omega$-limit viscosity solution $u$
is independent of $t$ on $\M=\RR\times \M_0$.
One can then conclude that the solution
$u$ is independent of $t$.

\medskip
The time-periodic case however is more complicated,
and we are not able to give a description
of $\omega$-limit orbits without using 
some specific features of the low dimension.
This will be done in the next section.

Before we continue, let us give an important remark.
All the objects constructed in this section, the 
sets $\D$ and $\M$ and the mappings $S^t$,
depend on the periodic solution $\phi$ that 
was chosen in the beginning.
Let us note $\D_{\phi}$ and $\M_{\phi}$ and 
$S^t_{\phi}$ in order to emphasize this dependence.
If $\psi$ is another $1$-periodic viscosity solution,
then we see from Corollary \ref{const} that
the orbits of $\M_{\psi}$ are calibrated by $\phi$.
It follows that the set 
$$
\A=\bigcap _{\phi}\D_{\phi},
$$
is not empty,
where the intersection is taken on the set
of $1$-periodic viscosity solutions.
This set is usually called the Aubry set.
The mappings $S^t_{\phi|\A_0}$
do not depend on $\phi$,
and for all $\phi$, we have 
$\M_{\phi}\subset \A$.



%
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\Section{Rotation number and convergence}
\label{circle}
%
%
%
%
%
%
In this section, we shall take advantage of the low dimension.
More precisely, we shall make use of Poincar\'e theory
of homeomorphisms of the circle, see \cite{KH} for example.
We have constructed in the previous section
a closed  subset $\A$ of $\RR\times \T$, 
which is the disjoint union of graphs of calibrated curves.
These calibrated curves will be called the orbits
of $\A$ from now on.
It will be useful to consider the standard projection
$\pi:\RR\times\RR \lto \RR\times \T$,
and the subset $\bar \A=\pi^{-1}(\A).$
Since $\A$ is the disjoint  union of the graphs of its orbits 
$\gamma:\RR\lto \T$, $\bar \A$ is the disjoint
union of graphs of continuous curves 
$\bar \gamma:\RR\lto \RR$.
These curves $\bar \gamma$ are called orbits of $\bar \A$,
they satisfy $\pi\circ \bar \gamma=\gamma$.
By straightforward extension of Poincar\'e Theory
of homeomorphisms of the circle,
we obtain the following results:

The limit 
$
\rho =\lim_{t\lto\infty} \bar \gamma(t)/t
$
exists and does not depend on the orbits $\bar \gamma$.
The number $\rho$ is called the rotation number.
The rotation number is rational  if and only if 
there exists a periodic orbit $\gamma(t)$ of $\A$.








\medskip
\noindent{\sc Proof of the theorem.} 
Let $u(t,x)$ be an $\omega$-limit
viscosity solution.
Let $\phi(t,x)=\liminf u(t+n,x)$
be as in Proposition \ref{liminf}.
We will prove that $u=\phi$ in the two cases 
$\rho=0$ and $\rho$ irrational.
The general case of a rational rotation number
$\rho=a/b$, $a\neq 0$  can be reduced to the
case $\rho =0$ by considering the Hamiltonian
$$
\tilde H(t,x,p)=aH(at,bx-at,\frac{p}{b})
$$
and noticing that the function $u(t,x)$
is a solution of the equation (\ref{HJ})
with Hamiltonian $H$ if and only if
the function $\tilde u(t,x)=u(at,bx-at)$
is a solution of the equation (\ref{HJ})
with Hamiltonian $\tilde H$,
and that the rotation number associated 
to this second equation is $\rho=0$.




In view of Lemma \ref{unique}, it is enough
to prove equality on $\M_{\phi}$.
Let us note $d=u-\phi$. 
This function is differentiable on $\M_{\phi}$
and satisfies 
$$\partial_td=\partial_xd=0.
$$
Recall that 
there exists a constant $K$ such that
the function $d(t,.)$ is $K$-Lipschitz
for each $t$.
Recall also that 
$\liminf d(n,x)=0$
for each $x\in \T$.

\noindent{\bf Case 1.} $\rho=0$.
In this case, the orbits $\gamma$ of $\A$ are of two kinds:\\
%
(I). Orbits of period $1$.\\
(II). Orbits heteroclinics between orbits of type (I).
This means that if $\gamma(t)$ is an orbit of type (II),
there exist orbits $\gamma^{+}$ and $\gamma^-$
of type (I) such that
$\lim_{t \lto \infty}(\gamma(t)-\gamma^{+}(t))=0$
and
$\lim_{t \lto -\infty}(\gamma(t)-\gamma^{-}(t))=0$.


It follows that the $\omega$-limit $\M$ is the union
of the graphs of the orbits of $\A$ of type (I).
Let $\gamma$ be such an orbit.
The function $d$ is constant on the graph of $\gamma$.
Since in addition we have 
$0=\liminf d(n,\gamma(0))=\liminf d(n,\gamma(n)$,
this constant is zero.
As a consequence, $d=0$ on $\M$.

\medskip
\noindent {\bf Case 2.} $\rho$ is irrational. 
In this case, there is no periodic orbit in $\A$.
We have to describe the complement of $\A$:
\begin{lem}
Each connected component $\bar U$
 of the  complement 
of $\bar \A$ in $\RR\times\RR$ is of the form
$$
\big\{
(t,x)  \hbox{ such that }
\bar \gamma^-(t)\leq x \leq \bar  \gamma^+(t)
\big\},$$
where $\bar \gamma^{\pm}(t)$ are two orbits of $\bar \A$
which satisfy $\lim |\bar \gamma^+(t)-\bar \gamma^-(t)|=0$.
\end{lem}
\noindent{\sc Proof.}
It is quite clear that each connected    component $\bar U$
 of the  complement 
of $\bar \A$ in $\RR\times\RR$ is of the form
$
\bar \gamma^-(t)\leq x \leq \bar  \gamma^+(t)
.$
In order to prove that
 $\lim |\bar \gamma^+(t)- \bar\gamma^-(t)|=0$,
let us consider the interval 
$\bar I_k=]\bar \gamma^-(k),\bar \gamma^+(k)[\subset \RR$.
We claim that the intervals $I_k=\pi(\bar I_k)$ are all disjoint
in $\T$.
Indeed, recalling that $\A_0$ is the set of points 
$x\in T$ such that $(0,x)\in \A$, we see that each of the
intervals $I_k$ is a connected component in $\T$ of the complement
of $\A_0$. 
Now suppose that $I_k$ and $I_l$ have nonempty intersection.
then the boundaries of $I_k  \cap  I_l$ are points
of $\A_0$, (since the boundaries of $I_k$ and $I_l$ are)
so they are contained neither in $I_k$ nor in $I_l$,
which is possible only if $I_k= I_l$.
Now assume that $I_k=I_l$ with $k < l$.
This implies that $\gamma^+(l)= \gamma^+(k)$,
hence $\gamma^+(t) $ is periodic, which is a contradiction.
So the intervals $I_k$ are all disjoint, hence their
lengh $\bar \gamma^+(k)-\bar \gamma^-(k)$ have finite sum
and so converge to zero.
\finproof


The proof that $d|_{\A}=0$ is similar to the proof
of Proposition 6.5. in \cite{Be}.
Let us set $\bar d=d\circ \pi$.
Let $\bar U$ be a connected component of  the
complement of $\A$, and let $\bar \gamma^{\pm}$
be its boundary curves.
Clearly, $\bar d(t,\bar \gamma^+(t))$ and 
 $\bar d(t,\bar \gamma^-(t))$ are constants
$c^+$ and $c^-$.
In addition, since $\liminf \bar \gamma^+ -\bar \gamma^-=0$
and since the functions $\bar d(t,.), t\in \RR$ are equilipschitz,
we have  $c^+=c^-$.
Since this holds for all connected component $\bar U$
of the complement, there exists a continuous function 
$D:\RR\times\RR\lto \RR$ which coincides with
$\bar d$ on $\bar \A$ and is constant on the complement
of $\bar \A$.
Since $d$ is flat on $\A$, the function $D$ clearly has the
same property, hence $D$ is differentiable at each point
with zero differential.
It follows that the function $D$ is constant,
hence $\bar d$ is constant on $\bar \A$, hence
$d$ is constant on $\A$.
Recalling that $\liminf d(n,x)=0$ for $x\in \A_0$,
we obtain that $d=0$ on $\A$.
\finproof



\bigskip
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\end{document}


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