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%% WS-P8-50x6-00.CLS : 20-11-97
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\end{filecontents}
\documentclass{ws-p8-50x6-00}
\input amssym.def
\input amssym.tex
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% TO The TeXnician:
%% The following two \defs
%% are hacks to obviate some incompatibilities
%% introduced misteriously
%% by the package of macros use.
%% What I would have liked is to have
%% \def\Dalambert{\square}
%% and leave \newblock alone.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\vrule \vbox{
\hrule
\hbox to 8 pt{
\vrule height 6 pt depth 2pt }
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}\nolimits}
\def\newblock{ }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\mycite#1{(\cite{#1})}
\begin{document}
\title{Variational methods for quasi-periodic solutions of
partial differential equations}
\author{Rafael de la Llave}
\address{ Mathematics Department, The University of Texas at Austin\\
Austin, Texas 78712-1082, USA \\
{\tt llave@math.utexas.edu}}
\maketitle
\abstracts{
We show how to use variational methods to prove
two different results: Existence of periodic
solutions with irrational periods
of some hyperbolic equations in one dimension
and existence of spatially quasi-periodic solutions
of some elliptic equations.
}
\section{Introduction}
In this note, we will present two results.
The first is a remark about a question of \mycite{Brezis83}
on whether one can use variational methods to produce solutions with
irrational period of
\begin{eqnarray*}
&{\partial^2u\over\partial t^2} - {\partial u\over\partial x^2} +V(u)=0\cr
&u(0,t) = u(1,t)
\end{eqnarray*}
We will show that for some particular numbers there is an extremely
simple answer.
The second result is to show how one
can use variational methods to produce
quasiperiodic solutions to some
difference equations as well as
partial differential equations
and to equations involving
some pseudo-differential operators.
The method of proof is related to some of
the results of Aubry-Mather theory in
dynamical systems and we recover the
corresponding results in dynamical
systems as particular results.
Of course, the Aubry-Mather theory
for dynamical systems
is much more developed and
it includes not only results
about existence of minimizing
periodic orbits, but also construction of
connecting orbits, etc. Many of these
results do not have an analogue yet for
PDE's. The connection of Aubry-Mather
theory with PDE's was pointed out in
\mycite{Moser86}. Results for
difference equations were obtained in
\mycite{Blank89}.
Here we will present two proofs for partial difference equations
that are based respectively in \mycite{KochLR97} and \mycite{CandelL98}.
These strategies can be adapted to the more complicated case of partial
differential equations.
As it will appear in the proof, the arguments rely just on translation
invariance, comparison principles, periodicity, and a variational structure
(for differential and pseudo-differential equations
we will also need some compactness properties, which
amount to regularity results)
and, hence they work in great generality.
For the purposes of these notes we will just present the simplest cases.
We point out that an important element in
our proof is the consideration of the heat flow as
a tool. The relevance of the heat flow
for Aubry-Mather theory seems to have originated in
\mycite{Angenent90} and was used in \mycite{Gole92}.
\section{Periodic solutions of irrational periods for wave equations}
\subsection{Statement of results}
We recall that a number $T\in\real$ is called of constant type if
$|Tk-l| \ge C|k|^{-1}$ $\forall\ k,l\in\integer$.
It is well known (Liouville's theorem)
that all irrational numbers which solve quadratic
equations with rational coefficients are of constant type.
We sketch a proof of the delightful classical argument.
In effect, if $\xi$ is an irrational number and $P(\xi)\equiv a\xi^2 + b\xi
+c =0$ with $a,b,c \in\integer$, we have $P'(\xi)\ne0$ because, otherwise
$P(x) = a(x-\xi)^2 = ax^2 - 2\xi ax + a\xi^2$, and, then $\xi = -b/2a$
would be rational.
If $n,m\in\integer$ and $|m/n-\xi|$ is sufficiently small, by the mean
value theorem
$$|P(n/m)| = |P(n/m) - P(\xi)| \le 2|P'(\xi)|\ |n/m-\xi|$$
On the other hand
$$|P(n/m)| = {1\over m^2} |an^2 + bnm + cm^2| \ge {1\over m^2}$$
because the numerator is an integer and different from zero
by hypothesis.
Putting together the two inequalities above,
we have:
$$|n/m -\xi| \ge {1\over 2|P'(\xi)|}\ {1\over m^2}.$$
\qed
Quadratic irrationals are not the only constant type numbers.
It is easy to show that a number is constant type if and only
if it has a bounded continued fraction expansion.
In particular, the set of constant type numbers is uncountable
(but it has measure zero).
As usual, given a function $f:\real\times\real\to\real$ satisfying
$f(x,t) = f(x+1,t) = f(x,t+T)$.
We will write
$$f(x,t) = \sum_{k,l\in\integer} \widehat f_{k,l}
e^{2\pi i[kx+l (t/T)]}$$
and denote
$$H^s = \biggl\{ f \big| \|f\|_s^2 = \sum_{k,l\in\integer}
|\widehat f_{k,l}|^2 (1+k^2 + l^2)^{s} <\infty\biggr\}\ .$$
We will also denote by $\Dalambert$ the operator
${\partial\over\partial t^2}-{\partial\over\partial x^2}$.
\begin{theorem}
\label{constanttype}
Let $T$ be a number of constant type.
Let $V:\real\to\real$ satisfy
\begin{itemize}
\item[i)] $0<\alpha\le V'\le \beta$,
where $\alpha$ is any number bigger than zero and $\beta$
depends on $T$ in a way that will be made explicit
\item[ii)] $|V''(x)|\le K$
\end{itemize}
Let $f:\real\times\real\to\real$ satisfy
\begin{itemize}
\item[iii)] $f(x+1,t) = f(x,t+T) = f(x,t)$
\item[iv)] $f\in L^2$
\end{itemize}
Then, there exist a $u\in L^2$ such that:
\begin{itemize}
\item[a)] $\Big( {\partial^2\over\partial x^2} -
{\partial^2\over \partial t^2}\Big) u+ V(u) = f$
\item[b)] $u(x+1,t) = u(x,t+T) = u(x,t)$
\end{itemize}
\end{theorem}
As we will see later, the smallness hypothesis on $\beta$ are
optimal.
Of course, $u$ satisfies a) only in the weak sense at this
stage (as it is typical of variational results).
For rational $T$, \mycite{Brezis83,BrezisN78} contain a
beautiful argument that shows that weak solutions are
strong solutions. This argument does not seem
to be available in our case. Nevertheless, we
can obtain smooth solutions for constant type
periods imposing smoothness in $V$ and $f$
and smallness in $f$.
\begin{theorem}
\label{unique}
Let $T$ be an irrational
constant type number as before.
Assume that $f\in H^r$ and $V\in C^{r+2}$ and that
its $C^{r+2}$ norm is sufficiently small.
Then, there is one $u \in H^r$ which satisfies
$$\Dalambert u + V(u) =f$$
The solution is unique in a ball in $H^r$ around the origin.
\end{theorem}
Note that, in particular, when $V(0)=0$, $f=0$, the only solution with
period $T$ is $u\equiv 0$ contained in a sufficiently small ball around
the origin.
This should be compared with the results of \mycite{CraigW93} which construct
solutions of $\Dalambert u+V(u)$ for many frequencies.
\subsection{Proof of Theorems \protect{\ref{constanttype}, \ref{unique}} }
The main observation is that, under the hypothesis that $T$ is a constant
type number $\Dalambert$ has closed range in $H^s$.
In effect, the range is just the closed subspace of functions $f$ such that
$\widehat f_{0,0} =0$.
Note that $\Dalambert$ is diagonal on trigonometric functions and that if
$f$ is such that $\widehat f_{0,0}=0$ if we denote by
$$\Dalambert^{-1} f
= \sum_{k,l} \widehat f_{k,l} {1\over (2\pi i)^2(-k^2+l^2/T^2)}
\ E_{k,l}(x,t)
$$
where $E_{k,l}(x,t) = \exp[ 2 \pi i (k x + l (t/T) ] $,
we have that on functions $f$ with zero average
$$
\Dalambert\, \Dalambert^{-1} f= \Dalambert^{-1}\, \Dalambert f = f .
$$
(Even if the first
can only be defined for functions with zero average,
we can define the second for
any $f$ and have $\Dalambert^{-1}\, \Dalambert f = f - {\hat f}_{0,0}$.)
The crucial remark
is that when $T$ is constant type
we have $$ |k^2 - (l/T)^2| \ge C > 0 \ \ \forall (k,l) \in \integer^2 - (0,0)$$.
Since
$$k^2 - {l^2\over T^2} = T^{-2} (Tk-l)(Tk+l)$$
we see that when $k,l$ are of the same sign, we can bound
$|Tk-l|\ge C|k|^{-1}$ and $|Tk+l| >T|k|$, therefore
the product can be bounded from below by $CT$.
When $k,l$ are of opposite sign,
$|Tk-l|\ge T|k|$ and $|Tk +l|>C |k|^{-1}$. \qed
This shows clearly that $\Dalambert^{-1}$ can be defined as
a bounded operator on the space of functions with zero average.
Since it is clear that the average of a function in the
range is zero, $ \Ran \Dalambert$ is precisely the
set of functions in $H^r$ with zero average. Clearly, it is
a closed subspace.
Note also that $\Ran (\Dalambert)^\bot = \{\hbox{constant functions}\}$.
The rest of the proof of Theorem \ref{constanttype} is very similar
to parts of the corresponding result in \mycite{Brezis83,BrezisN78}, which we
reproduce for the sake of completeness. Unfortunately, the
bootstrap of regularity in these references, does not go through in
our case, indeed, we suspect that the conclusions of the bootstrap
may not be true in our case.
Note that $\Dalambert u+V(u)=f$ implies that $V(u)-f\in \Ran (\Dalambert)$
and that, since $0<\alpha\le V'\le\beta$ we can find $V^{-1}$ and
that $\beta^{-1}\le (V^{-1})' \le\alpha^{-1}$.
If we write $V(u) -f=w$, $u= V^{-1} (w-f)$ the original problem becomes
$$V^{-1} (w-f) -\frac1T \int V^{-1} (w-f) +\Dalambert^{-1}w=0$$
where $w\in\Ran(\Dalambert)$, that is
$$V^{-1} (w-f) + \Dalambert^{-1} w\in \Ran (\Dalambert)^\bot
\label{eq:1}
$$
This problem has a variational structure.
If $\V$ is a primitive of $V^{-1}$, then (\ref{eq:1}) is the equation for the
critical points of
$$\SS(w) = \int \V (w-f) + (1/2) (w,\Dalambert^{-1} w)$$
considered as a functional on the space $Y=\Ran (\Dalambert)$.
Since $Y$ is closed, it is a Banach space.
Now we turn to computing the second derivative of
the functional.
We leave to
the reader the easy verification that the functional admits second
Gateaux derivatives and that they are
what one would expect through a formal
calculation. The only non-trivial part is to check
that $w \to \int \V (w-f)$ is differentiable, which follows
easily
from the Moser estimates for Sobolev spaces (See \mycite{Taylor97} 13 \S 3.)
We have that:
$$D^2 \SS (w)\gamma = \int \V'' (w-f)\gamma^2 + (\gamma,\Dalambert^{-1}
\gamma)$$
Since the second derivative of $\V$ is the first derivative of $V^{-1}$
we see that
$$\int \V'' (w-f)\gamma^2 \ge \beta^{-1} T\|\gamma\|_{H_0}^2$$
Since $(\gamma,\Dalambert^{-1}\gamma) \le T^{-1} C \|\gamma\|_{H_0}^2$
if $$\beta^{-1} T>T^{-1} C4\pi^2$$, the second derivative of
the functional $\SS$ is strictly
positive definite and, therefore the functional is strictly convex.
The condition above is the condition on $\beta$ that we alluded to in the
statement of the theorem. Note that it involves $T$ and
its number theoretic properties.
If this condition holds $\SS$ is a convex functional, moreover, we have
\begin{equation}
\label{eq:2}
\SS(w) \ge a\|w\|_{H_0}^2 - b\|w\|_{H_0} -c
\end{equation}
for some $a,b,c \in \real^+$.
The functional is bounded from below and we can find a sequence $w_n$
such that $\SS(w_n)\to \inf_{w\in\Ran(\Dalambert)} \SS(w)$.
The bounds (\ref{eq:2}) show that $\|w_n\|_{H_0}$ is bounded,
hence we can extract a weakly convergent subsequence by Banach-Alaoglu
theorem.
As it is well known, convex differentiable functionals in Hilbert spaces
are lower semi-continuous in the weak topology.
Hence, the infimum is reached by the limit point and this limit point
satisfies the variational equations.
This finishes the proof of Theorem \ref{constanttype}.
\qed
Note that the conditions in $\beta$ we obtained are optimal
for theorems of the form indicated.
If we consider $V(u)=\lambda u$, and $\lambda$ is an eigenvalue of
$\Dalambert$, we do not expect any solutions.
The eigenvalues of $\Dalambert$ get as close as desired to the value we claimed.
The proof of Theorem \ref{unique} is quite elementary.
If we write $u= u_1 +\delta$, where $u\in\Ran (\Dalambert)$, $\delta\in\real
= \Ran (\Dalambert)^\bot$, the equation $\Dalambert u+V(u)=f$ is equivalent to
\begin{eqnarray*}
u_1 & = -\Dalambert^{-1} [V(u_1+\delta) -f)\cr
\delta & = \frac1T \int [V(u_1+\delta)-f]
\end{eqnarray*}
We only have to check that the right hand side
of the equation is a contradiction
if we consider it as an operator acting on $(u_1,\delta)$ with a
contradiction constant that can be controlled by $\|V\|_{C^{r+2}}$.
This follows from the Moser estimates standard in
Sobolev spaces. See e.g. \mycite{Taylor97} 13 \S 3)
\qed
\section{Aubry-Mather theory for configurations on lattices}
The classical Aubry-Mather theory is concerned with variational problems
for functions defined on the integers.
A prototype example is the variational principle
$$L(u) = \sum_{i\in\integer} (1/2) (u_i - u_{i+1})^2 - S(u_i)$$
with $S(u+1) = S(u)$.
The Euler equations become
\begin{equation}
\label{eq:3}
-u_{i+1} - u_{i-1} + 2u_i - S'(u_i) =0
\end{equation}
which is the well known ``standard map'' of dynamical systems.
Notice that, besides being a variational principle for the standard map,
we could consider $L(u)$ as the energy of a chain of penduli coupled by
springs. The $u_i$ would be the angle between the $i^{th}$ pendulum and the
rest position, $S(u_i)$ the potential
energy due to the position of the pendulum.
Another alternative physical interpretation (which appeared in
solid state physics) is a chain of atoms in a one dimensional
periodic potential coupled by a harmonic interaction.
With the latter physical interpretation, this is called
the Frenkel-Kontorova model, which is a qualitative model
of deposition on the surface of (one-dimensional) crystals.
The goal in this lecture is to investigate the existence of quasi-periodic
solutions of similar equations when the independent variables are in
$\integer^d$ not just in $\integer$.
This could be interpreted physically as arrays of penduli coupled to their
nearest neighbors.
That is, we want to consider the variational principle for maps
$u:\integer^d \to \real$
$$L(u) = \sum_{|i-j|=1} (1/2) (u_i-u_j)^2 - \sum_i S(u_i)$$
and its variational equation
\begin{equation}
\label{eq:4}
(-\Delta u)_i - S'(u_i) =0
\end{equation}
where $\Delta$ is the discrete Laplacian $(\Delta u)_i = \sum_{|j-i|=1}
u_j - 2du_i$ and, again $S(u+1) = S(u)$ which we will assume is a $C^2$
function
corresponding to the position and $(1/2)(u_i - u_{i+1})^2$ the elastic
energy of the spring between two consecutive penduli.
One of the first results of Aubry-Mather theory is that, for every
$\omega \in\real$, (\ref{eq:3}) has a solution $u_i$ such that
$\sup_i |u_i-\omega i|<\infty$.
This result is somewhat surprising since for the apparently very similar
equation $u_{i+1} + u_{i-1} - 2u_i = a$ with $a\ne 0$, the
solutions grow parabolically.
We refer to \mycite{MatherF91} for an up-to-date review of Aubry-Mather theory that,
nowadays, includes not only the existence of solutions as indicated above,
but much deeper geometric characterizations of the quasi-periodic invariant
sets, their stable manifolds and many other geometric
and dynamical properties of these sets.
We will present two proofs of the following result. The
proofs are based on \mycite{KochLR97,CandelL98}.
There, the reader will find more details as
well as generalizations. Another proof can be found
in \mycite{Blank90}.
\begin{theorem}
\label{lattice}
With the notations above, for every $\omega \in\real^d$ there exists a
solution of (\ref{eq:4}) such that
$$\sup_i |u_i - \langle \omega,i\rangle| <\infty$$
\end{theorem}
The proofs we will present make use of the heat flow, whose importance
was noted in \mycite{Angenent90,Gole92}.
Our goal here is to present the argument in its simplest form so that it
is clear that it is a consequence of only:
\begin{itemize}
\item Variational structure
\item The variational structure is somewhat local
\item Periodicity in space of the variational problem
\item Periodicity in the $u$'s of the variational problem
\item Twist conditions (comparison theorems for the gradient flow)
\item Some mild regularity assumptions that guarantee the existence of
a flow for all times as well as differentiability of
functionals
\end{itemize}
As detailed in \mycite{KochLR97,CandelL98}
there are several other contexts
where we have the properties above and, hence we have an analogue of
Theorem \ref{lattice}.
As for the locality of the variational principle,
we point out that it is very mild. It will be
apparent in the details of the proof.
The locality requirements in the proof in
\mycite{KochLR97} are slightly stronger than those in
\mycite{CandelL98}.
We again note that the variational structure of the problem
with periodicity conditions is crucial since
the equation $-\Delta u+a=0$ leads to quadratic growth at $\infty$.
{From} the point of view of variational theory these are somewhat non-standard
problems since the spaces of trial functions are modeled in the spaces
$\ell^\infty (\integer^d)$ and, the functionals are only formal --- the sums are
not meant to converge.
If we try to regularize the problem by well defined ones, we see that
we have to get some control at $\infty$ to guarantee the linear growth.
This is conveniently achieved by introducing the customarily called
Birkhoff configurations.
\begin{defi}\label{Birkhoff}
We say that $u$ is a Birkhoff configuration if for every $k\in\integer^d$,
$l\in\integer$ we have either
$$u_{i+k} +l\ge u_i\ \forall\ i\in\integer^d\qquad\hbox{ or}\qquad
u_{i+k} +l\le u_i\ \forall\ i\in\integer^d$$
\end{defi}
That is, if we translate the graph horizontally and vertically by
integers, the graph does not cross itself.
Given $\omega\in \real^d$ we denote
$$\B_\omega = \{u\mid u\hbox{ is Birkhoff }u_i
-\langle \omega,i\rangle\in\ell^\infty\}\ .$$
We note that $\B_\omega$ is not empty since $\langle \omega,i\rangle$ belongs
to it.
We also note that if $u\in \B_\omega$ then $u_k-u_0$ is restricted to lie
in an interval of length~1.
In effect, if $u_k -u_0+l \ge0$ then because of the Birkhoff property
$u_{nk+k} - u_{nk}+l\ge0$ and adding we obtain
$u_{(n+1)k} - u_0 + n l \ge0$ (similarly for $\le$).
Since $u_{(n+1)k} -\langle \omega,(n+1)k\rangle$ is to remain bounded
independently of $n$ we see that the inequality we have to pick in
Definition \ref{Birkhoff}
is the same as that when we compare $\langle \omega,k\rangle$
with $l$.
For $k\in\integer^d$, $l\in \real$ we introduce
$(\C^ku)_i = u_{i+k}$;
$\R^l u_i = u_i+l$.
Note that $\C^k \C^j = \C^{k+j}$, $\R^l \R^n = \R^{l+n}$
and $\C^k \R^l = \R^l \C^k$.
A more compact notation can be obtained as follows:
We say that $u\ge v\Leftrightarrow u_i\ge v_i$ for all $i$
(similarly for $\le$)
[note that this is not a total order, there are many non-comparable elements]
Using this notation, we
can say that a configuration $u$ is Birkhoff if and only if, for
every $k\in\integer^d$, $l\in\integer$
$$\C^k\R^l u\ge u\qquad\hbox{ or }\qquad
\C^k \R^l u \le u\ .$$
This makes it clear that
$$\C^k \B_\omega = \R^l \B_\omega = \B_\omega$$
We also note that $\B_\omega$ is closed under pointwise limits.
It will be important to note that the
fact that a configuration is Birkhoff
implies a priori bounds for the components.
If $u \in B_\omega$, then $u_{k+i} + l - u_i$
has to have the same sign as $\langle \omega , k \rangle + l$.
(Otherwise, applying the Birkhoff property, we
would get a contradiction with
$u_i -\langle \omega , k \rangle \in \ell^\infty$ )
Therefore, evaluating for $i = 0$ we obtain that
when $u \in \B_\omega$,
$u_k - u_0$ has to lie on an interval of length
two (which we can write explicitly as a function of $k$).
Hence, if we consider the subset of $u \in \B_\omega$ with
$u_0 \in [a,b]$, applying Tychonov theorem, we conclude
it is compact under the pointwise limit topology
and, since it is closed, we obtain it is compact.
Relatedly, if we consider two configurations as equivalent if they differ
by an integer --- this is natural since this does not affect the variational
principle --- we can choose a representation with
$u_0\in \torus^1 \equiv \real/\integer$.
Hence,
$$\B_\omega /\R\ \hbox{ is compact.}$$
The next ingredient in the proofs is the heat flow.
This is the solution of the equation
$$\frac{\partial u}{\partial t} = \Delta u- S'(u)$$
Note that, formally, this is the gradient flow for the (strictly speaking,
meaningless) variational principle.
There is no problem in showing that the heat flow is defined for all time
since the R.H.S. of the equation is Lipschitz on
$\ell^\infty(\integer^d)$.
We denote by $\Phi_t(u)$ the solution at time $t$ of initial condition $u$.
An important result for the proof is the comparison principle
\begin{lemma}
If $u\ge v$ then $\Phi_t(u) \ge \Phi_t (v)$.
\end{lemma}
The proof of this lemma is not difficult.
The crucial observation is that the
twist condition implies that
the off-diagonal terms in the
variational equations of the flow are non-negative.
A simple calculation shows that the
elements of the matrix of the variational
equation are:
$ - \frac{\partial^2 L(u)}{\partial u_i\partial u_j} $.
The non-negativity of this quantity is the
twist hypothesis.
Now we observe that
comparison is true for linear equations with diagonal coefficients
(they become exponentials) and for those who have positive entries ---
use the Taylor expansion of the exponential.
Using Trotter product formula, it is true for sums of diagonal and
positive entries, hence for the variational equations of heat flow.
Once the result is true for the variational equations it is true for
the full equations by interpolation.
We refer to \mycite{KochLR97} for more details if needed.
\qed
We also note that $\R^l \C^k \Phi_\tau = \Phi_\tau \C^k \R^l$.
Hence, the heat flow of a Birkhoff configuration is Birkhoff.
Note also that if $u\in\langle \omega,i\rangle +\ell^\infty$ then
$\Delta (u) + S(u)\in \ell^\infty$, therefore
$$\Phi_\tau \B_\omega \subset \B_\omega$$
and, moreover, the heat flow projects to $\B_\omega/\R$.
As remarked before, the heat flow is the gradient flow of the variational
principle. But recall that since $L(u)=L(\R^l u)$ the variational
principle is ``defined'' on the compact set $\B_\omega/\R$.
Hence, the gradient should vanish at one point in the set.
Unfortunately, the previous argument does not work since the variational
principle $L$ is not ``defined'' in anything except a formal sense.
So we need to regularize somehow and study the convergence of the
regularizations.
The first solution that we present is that in \mycite{KochLR97}.
For a cube $\Lambda$ we define:
$$L_\Lambda (u) = \sum_{\scriptstyle i\in\Lambda\atop\scriptstyle |i-j|=1}
(1/2) |u_i - u_j|^2 - \sum_{i\in\Lambda} S(u_i)$$
We note that $L_\Lambda (u)$ is a well defined function in $\B_\omega/\R$
continuous when $\B_\omega/\R$ is given the pointwise convergence topology.
Moreover we have:
\begin{equation}
\label{eq:5}
\frac{d}{dt} L_\Lambda (\Phi_c u) = -\sum_{i\in\Lambda} (-\Delta -S'(u))^2
+ \sum_{\scriptstyle i\in\partial\Lambda\atop\scriptstyle |i-j|=1} J_i(u)
\end{equation}
where the $J_i$ are ``gradients.''
The intuition is that for large enough cubes the boundary terms should be
ignorable. A moment's reflection shows that this should be the case,
except when the bulk terms $\sum_{i\in\Delta}$ are very small, but in
this case, we should have a critical point!
More precisely,
\begin{lemma}
\label{lem:2}
Assume that there is no solution of (\ref{eq:4}) in $\B_\omega$ then, we
can find $\varepsilon >0$, $N$ such that if $u\in \B_\omega$ $|\Lambda|>N$
then $\sum_{i\in\Lambda} (-\Delta u+S'(u))^2\ge\varepsilon$.
\end{lemma}
We show that the negation of the conclusion implies the negation of the
assumptions.
The negation of the conclusions is:
\begin{itemize}
\item $\exists \{u^{(n)}\}_{n=0}^\infty, \{\Lambda_n\}_{n=0}^\infty$
such that
$u^{(n)} \in \B_\omega$, $|\Lambda_n|\to\infty$,
\item $\Gamma_{\Lambda_n}(u^{(n)}) \equiv \sum_{i\in\Lambda_n}
[-\Delta u^{(n)} + S'(u^{(n)})]^2\to\infty$.
\end{itemize}
In such a case,
we can find $k_n$ such that $\widetilde\Lambda_n = \C^{k_n} \Lambda_n$
is centered at the origin.
Since
$$\Gamma_{\C^{k_n}\Lambda_n} (\C^{k_n} u^{(n)}) = \Gamma_{\Lambda_n} (u^{(n)})$$
we see that the sequence $\C^{k_n} u^{(n)}$ --- consisting of configurations
in $\B_\omega$ --- makes arbitrarily small.
Hence $\Gamma_{\tilde\Lambda_n} (\tilde u^n) \to0$ where
$\widetilde \Lambda_n \subset \widetilde \Lambda_{n+1}$.
By passing to a subsequence (the Tychonov topology is metrizable in this case)
we can assume that $\tilde u^{(n)} \to u^*$.
Since $\Gamma_{\tilde\Lambda_{-(n+m)}} (u)\ge \Gamma_{\tilde\Lambda_n}(u)$
for all $u$, we have
$$\Gamma_{\Lambda_n} (\tilde u^{(n+m)}) \mathop{\longrightarrow}\limits_{m\to
\infty} 0$$
and therefore $\Gamma_{\Lambda_n} (u^*) =0$ for all $n$.
Hence $u^*$ satisfies (\ref{eq:4}) \qed
\begin{corollary}
In the assumptions of Lemma \ref{lem:2} we can find $C>0$ such that
$$\Gamma_\Lambda (u) \ge C(|\Lambda| -1)\ \forall \ u\in \B_\omega$$
\end{corollary}
Given a large enough cube, divide it in cubes of size $N$ and apply to
each of the cubes Lemma \ref{lem:2}.\qed
It is very easy to show that the terms $J_i$ are uniformly bounded, hence
if there was no critical point, applying Corollary~1 to estimate the
right hand side of (\ref{eq:5}) we have:
$${d\over dt} L_\Lambda \Phi_\tau (u) \le -C\Bigl[(|\Lambda|-1)
+ |\Lambda|^{1-(1/d)}\Bigr] \ \forall\ u\in \B_\omega$$
Hence, for big enough $|\Lambda|$, the right hand side would be negative.
But this is impossible since $L_\Lambda$ is a continuous function in the
compact set $\B_\omega/\R$.
We, therefore conclude that there is a critical point.
A second proof of the result of Theorem \ref{lattice} can be obtained by observing
that if $\omega = (1/N) (n_1,\ldots,n_d)$ with $N\in\natural$,
$n_1,\ldots,n_d\in\integer$ we can define
\begin{equation} \label{LN}
L_N(u) = \sum_{\scriptstyle |i-j|=1\atop\scriptstyle |i|_\infty\le N}
(1/2) |u_i - u_j|^2 - \sum_{|i|\le N} S(u_i)
\end{equation}
In this case, the Birkhoff orbits are periodic and satisfy:
$$u_{i_1,\ldots, i_j +N,\ldots,i_d}
= u_{i_1,\ldots,i_j,\ldots,i_d} + n_j$$
This implies that
$$\frac{d}{dt} L_N (u) = -\sum_{|i|_\infty\le N} (-\Delta u+ S' (u))_i^2$$
Hence, we can conclude that the heat flow leads to a solution of
\ref{eq:4} when $\omega \in \rational^d$.
To prove the result for any $\omega$ we observe that if we find a sequence
$\omega^{(n)} \to\omega$ with $u^{(n)} \in \B_{\omega^n}$ solving
\ref{eq:4}, (we can assume without loss of generality $|u_0^{(n)}| \le1$).
By the fact that $u^{(n)}$ in in $\B_\omega$ for every $j\in\integer^d$ we
know that $u_j^{(n)}$ lies in an interval of size
$|j|\max_n (\omega^{(n)},\omega)$
that is, it remains bounded.
Hence, by passing to a subsequence, we can obtain that the
$u^{(n)}\to\tilde u$ pointwise.
Since the $u^{(n)}$'s satisfy (\ref{eq:4}) so does $\tilde u$.
\begin{remark}
The proofs above do not need that the interaction is
nearest neighbor.
They go just through without changes
assuming
\begin{enumerate}
\item
\begin{equation}
L(u) = \sum_{B \subset \integer^d} H_B(u)
\label{interaction}
\end{equation}
where $H_B(u)$ is a $C^2$ function
which depends only on $u|_B$ and the sum is
formal.
\item The twist condition:
${ \partial^2 H_B(u) \over \partial u_i \partial u_j} \ge 0 $
when $i \ne j$.
\item Translation invariance:
$ H_{\R_k B}( \R_k u) = H_B(u) \ \
\forall u, \forall B \subset \integer^d, k \in \integer^d$.
\item Periodicity:
$H_B( \C_l u) = H_B(u)\ \
\forall u, \forall B \subset \integer^d, l \in \integer$.
\item Some conditions of decay for $H_B$ when $B$ is
large.
\end{enumerate}
Note that (\ref{interaction} ) is quite common in
Statistical Mechanics. $H_B$ describes the interaction
between the bodies in $B$. The conditions
of translation invariance appear very frequently in
Statistical Mechanics since it is a reflection
that the interactions do not depend on the translations.
The twist condition appears in statistical mechanics as
ferromagnetism.
As for the condition on decay, we will just say that it
is easy to check that the conditions are satisfied
when the interaction is finite range
(i.e. $H_B \equiv 0$ when the diameter of $B$ is large enough. )
Once we prove the result for
$
L_A (u) = \sum_{B \subset \integer^d, {\rm diam} B \le A } H_B(u)
$
we can consider taking the limit $A \to \infty $.
In order to do that, roughly, we need that $\nabla L(u)$
$A$ admits bounds independent of $A$ in all Birkhoff configurations.
The conditions of decay alluded in the last point are enough to
guarantee that. Note that in this argument, we do not need that
the heat flow is differentiable uniformly in $A$ since we
only take the limit of the solutions of the equilibrium
equations.
We refer to \mycite{KochLR97,CandelL98} for further details
and for generalizations.
\end{remark}
\begin{remark}
One can also generalize the results to other graphs that are
not just lattices.
The generalization of $\R_k$ is that there is a
group $G$ acting effectively. The analogue of
quasiperiodic solutions are going to be the
characters of the group.
Note that the first proof requires that, when we increase the size,
the boundary elements become negligible.
The second proof requires an analogue of the rational vectors.
This can be achieved for the ``residually finite'' groups.
(The whole lattice can be exhausted by considering larger
and larger cells which are the quotient of the group
by a subgroup of finite index).
This allows to consider some graphs such as the
homogeneous branching tree (called the Bethe lattice in
Statistical Mechanics)
\end{remark}
The following remark is based on
unpublished research by the author.
\begin{remark}
The second proof presented above generalizes
to partial differential equations.
The conditions of translation invariance,
for the variational principle
is quite straightforward to generalize.
Of course, to study the gradient flow
one has to rely on PDE methods
(parabolic regularity).
The analogue of the twist condition is that the
heat flow satisfies a comparison principle.
To justify passages to the limit etc.
we need something that shows that the
equilibrium configurations are equicontinuous
in small enough scales.
(The equicontinuity in the large is
still obtained from considering Birkhoff configurations
as before.)
For PDE's this can be obtained using
elliptic regularity theory.
Some equations to which the the above
conditions apply are
$$
\Laplacian u + V'(x,u) = 0
$$
where $V(x + e , u + l) = V(x,u)$
$\forall x\in \real^d, u \in \real, e \in \integer^d,
l \in \integer$
$$
\sum_{i = 1}^k L_i^2 + V'(x,u) = 0
$$
where $L_i$ are $\integer^d$ periodic vector fields
satisfying H\"ormander's hypoellipticity conditions
and $V $ is in the previous case.
\begin{equation}
\label{squareroot}
(-\Laplacian)^{1/2} u + V'(x,u)
\end{equation}
with $V$ as above.
We note that the last equation
is not a PDE, since
$(-\Laplacian)^{1/2}$ is a non-local operator.
Nevertheless all the argument can be carried though.
The comparison principle for the
heat flow follows because
$ e^{ t (-\Laplacian)^{1/2}}$ is convolution
with the Poisson kernel, as it is
well known in harmonic analysis, which is positive.
Then, one can use Trotter product formula as indicated
in the proof. (An alternative, more general, argument, suggested
by C. Gutierrez is to use the {\sl subordination } identity.)
Problems involving $(-\Laplacian)^{1/2}$ similar
to (\ref{squareroot}) appear in several applications.
Let us just note that they appear in the study
of geostrophic flows and in Levy diffusions.
One can also combine this treatment of PDE's with the previous
remarks and obtain quasiperiodic solutions in other
homogeneous spaces besides the torus. We need to assume that the
fundamental group satisfies the condition of being
residually finite.
\end{remark}
\section{acknowledgements}
The research of the author has been supported by NSF
grants. Section 3 is based on joint work with
A. Candel, H. Koch and C. Radin. Both D. Bambusi and
an anonymous referee made suggestions which improved
the exposition.
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\end{document}