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\title[Generalized Frenkel-Kontorova models]{
Ground states and critical 
points for
generalized Frenkel-Kontorova models}

\author{Rafael de la Llave}
\address{Department of Mathematics\\
University of Texas at Austin\\
Austin TX 78712-0257
}
\email{llave@math.utexas.edu}


\author{Enrico Valdinoci}
\address{ Universit\`a di Roma Tor Vergata
Dipartimento di Matematica
Roma, Italia}
\email{valdinoc@mat.uniroma2.it}

\subjclass[2000]{82B20,
70H12,
37A60, 
82C20, 
82C22, 
49J35}

\keywords{Minimal and nonminimal equilibrium
solutions
in statistical mechanics, variational methods,
Aubry-Mather theory}

\thanks{The work of RdlL was supported by 
NSF grants.
The work of EV was   
supported by MIUR Variational
Methods and
Nonlinear Differential Equations.}

\begin{document}


\begin{abstract}
We consider a multidimensional
model of
Frenkel-Kontorova type but we allow 
non-nearest neighbor interactions. 

For every possible frequancy vector, we 
show that there are 
quasi-periodic 
ground states which enjoy further geometric properties.
The gound states we produce are either bigger or
smaller than the state. They are are at bounded distance from
the plane wave with the given frequency. 

The comparison property above implies that the 
ground states and the translations are organized into 
laminations. If these leave a gap, we show 
that there are critical points inside the gap which 
also satisfy the comparizon properties. 

In particular, given any frequency, we show 
that either there is a continuous parameter of 
ground states or there is a ground state and 
another critical point which is not a ground state. 
\end{abstract}

\maketitle

\section*{Introduction}
This paper deals with existence and
multiplicity results for periodic and
quasiperiodic  equilibria 
of generalized Frenkel-Kontorova models.

The equilibrium solutions we construct have further 
geometric properties, such as satisfying 
comparison properties and lying at 
a bounded distance from a linear function. 
The results presented here generalize 
some well known results in Aubry-Mather 
theory. We allow the lattices to be of 
{\em any dimension} and we also allow 
interactions which are {\em not nearest 
neighbor}. We also allow {\em Degenerate twists}. 

Our Theorem~\ref{minimizersexist}
constructing minimizers generalizes
the main result in~[A83, AD83, M82], 
whereas our
Theorem~\ref{MAIN:TH} constructing 
solutions which are not minimizers is an
extension of the main result
of~[Ma86].  Our results and techniques
may be also applied to
more general lattices and group actions 
(see~[LlV06a, LlV06b] for a more abstract framework). 
Arguments along the same line 
as 
the ones presented here apply to 
PDEs too but, then, one has to pay
attention to regularity, existence 
etc., which are easier issues in our case (see~[LlV05]). 

The Aubry-Mather theory had its origin 
in~[A83, AD83, M82] and
important precedents may be considered the works
of~[M24, H32, P74, P79].
A deep connection with the
theory of viscosity solutions in PDEs
has been developed too
(see, e.g.,~[F97]).
The techniques we use here will be
a combination of the dynamical system and the PDE
approaches.

The situation we consider deals with a lattice 
$\integer^d$. At every site, we have a particle 
whose state can be described by one real parameter. 

A configuration is an asignation of a state to 
each of the sites. That is, a configuration is 
$u_i\in \real$, with~$i\in\integer^d$.

Given a configuration, 
we consider the formal variational principle
$$ S(u):= \sum_{i,j\in\integer^d} a_{ij}|u_i-
u_j-c_{ij}|^{p_{ij}} +\sum_{i\in \integer^d}V(u_i)\,.$$
We suppose that~$p_{ij}=\bar p(|i-j|)$, 
$c_{ij}=\bar c(|i-j|)$,
$a_{ij}:=\bar a(
|i-j|)$, with~$2\leq\bar p\le\Lambda$,
$-\Lambda\le\bar c\le\Lambda$,
$0\leq \bar a \leq \Lambda$ and~$\bar a(r)
=0$ if~$r \in\{0\}\cup[R,+\infty)$, for suitable~$\Lambda$, $R>0$.
The last assumption means that the interaction is finite range. 
A discussion of how one can extend our results by 
considering infinite range interactions as 
the limit of finite range ones is in [LlV06a].



We also suppose that~$V$ is smooth and that~$V(r+1)=V(r)$
for any~$r\in\real$.

{F}rom the physical point of view, such a model
may be seen as a system of (possibly nonlinear
and nonhomogeneus) interacting springs in a 
background periodic potential. Another possible 
physical interpretation is spin waves. 
It is a generalization of the so called
Frenkel-Kontorova model. Notice that
we allow lattices 
of higher dimensions and the interactions to 
be nonnearest neighbor, which is very natural 
for spin-waves. For simplicity, we require only 
finite range. The form of the interaction 
-- chosen here to be (possibly nonlinear)
harmonic oscillators -- is 
not too relevant. The only properties 
that matter are some convexity properties
(configurations tend to be aligned), and the fact that
the interaction
is coercive (it penalizes close particles for becoming very 
different). 
See~[LlV06a]. 

{F}rom the mathematical viewpoint, the formal functional~$S$
may be seen as a discretization  of analogous ones arising
in the calculus of variations in relation with degenerate 
elliptic~PDEs.
The degeneration, which corresponds to the case~$p_{ij}>2$,
then reflects into the flatness of the twist condition
in the dynamical system framework.

The following definitions are standard in the calculus of 
variations. 

We say that
a configuration~$u_i$ is a \emph{ground state}
if
\begin{equation}\label{GS}
\begin{split}  
\sum_{{i,j\in\integer^d}\atop{\min\{|i|,|j|\}\leq K}}& a_{ij}|u_i-
u_j-c_{ij}|^{p_{ij}} 
+\sum_{{i\in \integer^d}\atop{|i|\leq K}}
V(u_i) \\
& 
\leq\;
\sum_{{i,j\in\integer^d}\atop{\min\{|i|,|j|\}\leq K}} a_{ij}|v_i-
v_j-c_{ij}|^{p_{ij}} +\sum_{{i\in \integer^d}\atop{|i|\leq K}}
V(v_i)
\,,
\end{split}\end{equation} 
for any configuration~$v_i$ such that~$v_j=u_j$
as long as~$|j|>K$, for any given~$K>0$.


We say that~$u_i$ is a \emph{critical point} if
\begin{equation} \label{CP}
\begin{split}
& \sum_{j\in\integer^d} a_{ij}p_{ij}
(u_i-u_j) \Big[
|u_i-
u_j-c_{ij}|^{p_{ij}-2} 
+
|u_j-
u_i-c_{ij}|^{p_{ij}-2}\Big] + V'(u_i)
\\=\;\,&
\sum_{j\in\integer^d} a_{ij}p_{ij}c_{ij}
\Big[
|u_i-
u_j-c_{ij}|^{p_{ij}-2} 
-
|u_j-
u_i-c_{ij}|^{p_{ij}-2}\Big]
\end{split}\end{equation} 
for any~$i\in \integer^d$
(of course, the case of ideal springs~$p_{ij}:=2$
and~$c_{ij}:=0$ is the easiest 
to visualize). 

Note  that the series in~\eqref{GS}
and~\eqref{CP} are just finite sums.
Roughly,~\eqref{GS} says that, for a ground state, 
any change of a finite number of sites increases the energy
(note that we sum the change of energy in the terms which are 
affected).
Equation~\eqref{CP}
is just the 
Euler-Lagrange equation of the functional~$S$.
In general, the energy functional~$S$ is
only a formal sum and may well not converge, 
but the definitions of ground state and critical point make sense
since they involve only finite sums.

Ground states are critical points, as it is
easily checked. 

The first result we present
deals with the existence of
ground states which lie at a bounded distance
from any given linear function. These ground 
states  have an interesting geometric 
property (known in the literature under the name
of Birkhoff or selfconforming see~\eqref{BP} below). We will interpret 
geometrically this 
property by saying that the ground states and their 
translations are organized in a lamination. 

\begin{theorem}\label{minimizersexist}
Given any vector~$\sigma\in\real^n$,
there exists
a 
ground state~$u_i$ such that 
\begin{equation}\label{BondUn}
|u_i -\sigma\cdot i| \,\le\,
2\,,\end{equation}
for any~$i\in\integer^d$.

Moreover, 
\begin{equation}\label{BP}
\begin{split}
&{\mbox{
if~$j \in\integer^d$ and~$s\in\integer$
are such  that~$\sigma\cdot j\geq s$,}} \\
&{\mbox{
we have
that~$u_{i+j}\ge u_i+s$ for any~$i \in\integer^d$.
}}\end{split}\end{equation}
\end{theorem}
We observe that~\eqref{BP}
implies that~$u_{i+j}=u_i$ for any~$j\in\integer^d\cap\sigma^\perp$,
and thus, for~$\sigma\in\rational^d$, it is just a periodicity 
condition. 
When~$
\sigma\in\real^d\setminus\rational^d$,
we obtain that the solutions are quasiperiodic since they 
are the limit of periodic solutions. Here we follow 
the notation common in variational method of calling quasiperiodic 
the limits in whatever sense of periodic solutions. In 
other areas of dynamics, 
one uses a different name depending on the different 
meanings of limit (e.g. pointwise, uniform etc. )
So, our solutions could be called quasiautomorphic 
in the language of topological dynamics. 

The property in~\eqref{BP} 
says that the integer translation of 
the graph are all comparable functions.
Furthermore, it is easy to see 
that the pointwise limit of these functions 
is also a ground state. 
We will call \emph{lamination} a set of comparable functions whose 
graphs in~$\integer^d\times(\real/\integer)$ 
are a closed set under pointwise limits. 
These definition is very similar to the usual definition of 
laminations in topology. The graphs of each of the functions is 
a sheet. Nevertheless, in our case, we will not assume 
that different sheets are disjoint. We allow 
that they can touch, even if they cannot cross. 
As we will see later, with an extra assumption on 
the interaction we will prove a maximum principle 
that shows that the graphs of comparable equilibria
have to be disjoint. 



The ground states satisfying~\eqref{BP} are
organized in laminations, hence it is natural 
to formulate the results for laminations. 
It can happen that the lamination 
covers~$\integer^d\times(\real/\integer)$ -- this is 
analogue of the case of existence of invariant circles 
for twist maps -- or that it contains gaps, which is 
an analogue of the minimizers being a Cantor set. 

Our next result shows that, in the case that a  lamination
of minimizers contains gaps, we can find a further critical 
point in the gap. 

\begin{theorem}\label{MAIN:TH}
Assume that the graphs of~$u^{(0)}_i$ 
and~$u^{(1)}_i$ belong
to a lamination
of ordered ground states 
as produced in Theorem~\ref{minimizersexist}. Suppose 
that~$ u^{(1)}_i \geq  u^{(0)}_i$ 
for any~$i\in\integer^d$ 
and that~$ u^{(1)}_{i_o} >  u^{(0)}_{i_o}$ for some~$i_o
\in \integer^d$.

Then,
there exists a critical point~$u_i$ satisfying~\eqref{BP},
different from~$u_i^{(0)}$ and~$u_i^{(1)}$,
such that~$ 
u^{(1)}_i \geq u_i \geq  u^{(0)}_i$ for any~$i\in
\integer^d$.
\end{theorem}

Under a further assumption, the result of Theorem~\ref{MAIN:TH}
may be sharpened by obtaining
a strict order property. For this, we say that
the interaction is {\em transitive} if given any~$i$, $j\in
\integer^d$ there exist~$i_1,\dots,i_\ell\in \integer^d$
in such a way that
$$ a_{i i_1}, \, a_{i_1 i_2},\,\dots,a_{i_\ell j} 
\,\neq\, 0\,.$$
Physically, this means that two sites are always connected
by a certain system of springs.
Then, Theorem~\ref{MAIN:TH}
takes the following stronger form:

\begin{corollary}\label{MAIN:CO}
Suppose that the interaction is transitive.
Assume that the graphs 
of~$u^{(0)}_i$
and~$u^{(1)}_i$ belong
to a  lamination
of ordered ground states
as produced in  Theorem~\ref{minimizersexist}. 
Suppose
that~$ u^{(1)}_i \geq  u^{(0)}_i$
for any~$i\in\integer^d$
and that~$ u^{(1)}_{i_o} >  u^{(0)}_{i_o}$ for some~$i_o
\in \integer^d$.

Then,
there exists a critical point~$u_i$ satisfying~\eqref{BP}
and such that~$ u^{(1)}_i > u_i >  u^{(0)}_i$ for any~$i\in
\integer^d$.
\end{corollary}

One of the tool of the proofs
of Theorem~\ref{MAIN:TH}
and Corollary~\ref{MAIN:CO}
will be a
comparison property (see Lemmata~\ref{weak:co} 
and~\ref{strong:co} below). 

The proofs of Theorems~\ref{minimizersexist}
and~\ref{MAIN:TH}
that we present here are
remarkably simple and they depend 
on very few elements of 
the problem. Thus, the ideas we expose
may be applied to quite general 
variational problems which satisfy some sort of 
``ferromagnetism'', ``coercivity''
and ``invariance under translations'' 
(see~[LlV06a, LlV06b]).

\section*{Comparison Principles}

We will deal
with the following differential equation:
\begin{equation} \label{Cauc}\left\{
\begin{matrix}
-\partial_t \Phi^t(u)_i
=
\displaystyle\sum_{j\in\integer^d} a_{ij}
p_{ij}(\Phi^t(u)_i-\Phi^t(u)_j) \Big[
|\Phi^t(u)_i-
\Phi^t(u)_j-c_{ij}|^{p_{ij}-2} \\ \qquad
+
|\Phi^t(u)_j-
\Phi^t(u)_i-c_{ij}|^{p_{ij}-2}\Big] + V'(\Phi^t(u)_i)
\\ \qquad+
\displaystyle\sum_{j\in\integer^d} a_{ij}p_{ij}c_{ij}
\Big[
|\Phi^t(u)_j-
\Phi^t(u)_i-c_{ij}|^{p_{ij}-2} \\ \qquad
-
|\Phi^t(u)_i-
\Phi^t(u)_j-c_{ij}|^{p_{ij}-2}\Big]
\qquad{\mbox{ for $t>0$,}}\\
\\ \Phi^0(u)_i = u_i\,.
\end{matrix}
\right.\end{equation}
Here, we will take~$t\in[ 0,T^\star]$, for some~$T^\star>0$.
The above Cauchy problem is the gradient flow
of~$S$.

If we fix a frequency $\omega$, we can consider 
the RHS defined on  the affine space consisting 
of functions $u$ such  $| u_i  - \omega \cdot i| $
is bounded uniformly in $i$. We denote this 
space as $X_\omega$. It is an affine space modelled
on $\ell^\infty(\integer^d)$. We note that 
when $p \ge 2$, the RHS is a $C^1$ mapping from 
$X_\omega$ to $\ell^\infty$. Furtehrmore, the Lipschitz 
constant is uniform on bounded sets. 
Therefore, applying the standard theory of 
existence and regularity of solutions of equations on 
Banach spaces (or Banach manifolds) (see e.g. [H80])
we can conclude that, when the initial data $u$ 
range over a bounded set, the flow is uniquely defined
for a time $T > 0$ and, moreover 
that the flow depends $C^1$ on the initial conditions. 
Furthermore, we can compute the derivative of 
the flow integrating the variational equations. 

Our next goal is to use the variational equations to 
show that there are Comparison Principles for 
the equation.
We will prove a weak one, Lemma~\ref{weak:co} and 
a strong one,  Lemma~\ref{strong:co} 
which requires an extra hypothesis. As we will see 
later, as  a consequence of the 
Comparison Principles, we will obtain that, in the situations 
we are interested, the flow can be extended for all times. 

The Comparison Principles we obtain are the following:

\begin{lemma}[Weak Comparison Principle]\label{weak:co}
Let~\eqref{Cauc} holds.
Suppose that~$ u_i \ge v_i$
for all~$i \in \integer^d$. Then~$\Phi^t (u)_i \ge \Phi^t(v)_i$
for any~$i \in \integer^d$
and any~$t\in[0,T^\star]$.
\end{lemma}

\begin{lemma}[Strong Comparison Principle]\label{strong:co}
Let the interaction be transitive.
Let the assumptions of Lemma~\ref{weak:co} hold.
If also~$u \not\equiv v$, 
then~$\Phi^t(u)_i > \Phi^t(v)_i$ for any~$i \in \integer^d$
and any~$t\in[0,T^\star]$.
\end{lemma}

As we will now see in the course of the proof, 
Lemma~\ref{weak:co}
uses only 
that the nondiagonal elements of
the variational equation are nonnegative,
while Lemma~\ref{strong:co}
exploits the fact that
the variational
equations have a positive kernel.

In order to prove Lemma~\ref{weak:co}
we argue as follows.
For~$\lambda\in[0,1]$,
let~$u^\lambda_i:=\lambda u_i+
(1-\lambda)v_i$ and differentiate~\eqref{Cauc}
to obtain that
\begin{equation} \label{rea7}
-\partial_t (\partial_\lambda \Phi^t(u^\lambda)_i)\,=\,
\delta_i^\lambda (t) \,\partial_\lambda \Phi^t(u^\lambda)_i-
\sum_{{j\in\integer^d}\atop{0<|i-j|\le R}}\mu_{ij}^\lambda (t)
\,\partial_\lambda \Phi^t(u^\lambda)_j\,,\end{equation}
for suitable~$\delta_i^\lambda (t)$
and~$\mu_{ij}^\lambda(t)$. The main property we use is that
\begin{eqnarray}\label{def:mu}
\mu_{ij}^\lambda(t)
&=&
a_{ij}p_{ij}(p_{ij}-1) \Big( 
|
\Phi^t(u^\lambda)_i -\Phi^t(u^\lambda)_j-c_{ij}
|^{p_{ij}-2}\\
&& +|
\Phi^t(u^\lambda)_j -\Phi^t(u^\lambda)_i-c_{ij}
|^{p_{ij}-2}
\Big)
\ge 0\,.\nonumber
\end{eqnarray} Let now
$$ C_i^\lambda(t):= e^{\int_0^t \delta_i^\lambda(s)\,ds} \,\partial_\lambda
\Phi^t(u^\lambda)_i\,.$$
Then, from~\eqref{rea7},
\begin{equation}\label{eqf} 
\partial_t C^\lambda_i (t)\,=\,
\sum_{{j\in\integer^d}\atop{0<|i-j|\le R}}\nu_{ij}^\lambda (t)
\,
C_j^\lambda(t)  \,,\end{equation}
with
$$ \nu_{ij}^\lambda (t)\,:=\,
e^{\int_0^t \delta_i^\lambda(s)\,ds}
e^{-\int_0^t \delta_j^\lambda(s)\,ds}
\mu_{ij}^\lambda (t)\,\ge\,0$$
and
\begin{equation}\label{102}
C_i^\lambda(0)=u_i-v_i\ge 0\,.\end{equation}
Therefore,~$C_i^\lambda(t)\ge 0$ and so~$\partial_\lambda\Phi^t
(u^\lambda )_i\ge 0$ (as can be proved, for small~$t>0$,
via  Picard approximation and, for any~$t\in (0,T^\star]$,
by iterating the ODE flow).
Accordingly,
\begin{equation}
\label{rea9}
\Phi^t(u)_i =
\Phi^t(u^1)_i\geq 
\Phi^t(u^\lambda)_i\geq
\Phi^t(u^0)_i=\Phi^t(v)_i\,,
\end{equation}
for any~$\lambda\in [0,1]$, 
which completes the proof of Lemma~\ref{weak:co}.
\medskip

Let us now deal with the 
proof of Lemma~\ref{strong:co}.
If
\begin{equation}\label{cista}
{\mbox{$\Phi^{t_o}(u)_{i_o}=\Phi^{t_o}(v)_{i_o}$ for 
some~$i_o\in\integer^d$ and
some~$t_o>0$,}}\end{equation} 
we gather from~\eqref{rea9} that~$\partial_\lambda
\Phi^{t_o}(u^\lambda)_{i_o}=0$, thence
\begin{equation}\label{ght6}
C_{i_o}^\lambda(t_o)=0\end{equation} 
for any~$\lambda\in(0,1)$.

Since~$C^\lambda_{i_o}(0)\ge0$ and the map~$t\mapsto C^\lambda_{i_o}
(t)$ is nondecreasing due to~\eqref{eqf},
we conclude that
\begin{equation}\label{100}
{\mbox{$C^\lambda_{i_o} (t)=0$ for any
$t\in [0,t_o]$ and any 
$\lambda\in (0,1)$.}}\end{equation}
In particular, \begin{equation}\label{101}0=
C_{i_o}^\lambda(0)=u_{i_o}-v_{i_o}\,.\end{equation}
Now, 
we prove that
\begin{equation}\label{step}
\begin{split}
&\qquad\qquad\qquad{\mbox{if 
\eqref{cista} holds,}}\\ &{\mbox{then
$\phi^\tau(u)_j=\phi^\tau(v)_j$ for any
$\tau\in[0,t_o]$ and any
$j\in \integer^d$
such that
$a_{i_o j}\neq 0$.
}}
\end{split}
\end{equation}
To confirm this, we first note that, up to replacing~$u_i$
and~$v_i$ with~$\Phi^\tau(u)_i$ and~$\Phi^\tau(v)_i$,
we may and do assume that~$\tau=0$
in~\eqref{step}. We also
observe that,
given~$j$ in such a way that~$u_j\neq v_j$,
there 
exist~$\lambda_j\in(0,1)$
in such a way 
that\begin{equation}
\label{edf5}
u^{\lambda_j}_{i_o}\neq u^{\lambda_j}_j\,.\end{equation}
Indeed, 
if~$u^{\lambda}_{i_o}= u^{\lambda}_j$ for any~$\lambda$,
we would have that
$$ 0=\partial_\lambda(u^{\lambda}_{i_o}-u^{\lambda}_{j})
=(u_{i_o}-v_{i_o})
-(u_j-v_j)=0-u_j+v_j\,,$$
due to~\eqref{101},
which is a contradiction.

Moreover,
from~\eqref{100}
and~\eqref{eqf},~$\nu_{i_o j}^\lambda (t) C^\lambda_j(t)=0$
for any~$j\in\integer^d$, any~$t\in [0,t_o]$
and any~$\lambda\in (0,1)$.
In particular, if, by contradiction,~$u_j\neq v_j$ for 
some~$j$ with~$a_{i_o j}\neq 0$, we would 
have
that
$$ 0=\nu_{i_o j}^{\lambda_j} (0)\, C^{\lambda_j}_j(0)=
\mu_{i_o j}^{\lambda_j}(0)\, (u_j-v_j)$$
and so~$\mu_{i_o j}^{\lambda_j}(0)=0$.
By~\eqref{def:mu}, this would 
give that~$u^{\lambda_j}_j=u^{\lambda_j}_{i_0}$,
in contradiction with~\eqref{edf5}. 
This says that~$u_j=v_j$
for any~$j$ as long as~$a_{i_o j}\neq 0$, yielding~\eqref{step}.

The proof of Lemma~\ref{strong:co}
is now ended by recursion, thanks to the fact that
the interaction is transitive. For this,
let~$t_o$ and~$i_o$ be as in~\eqref{cista}, take any~$j\in\integer^d$
and suppose that
$$ a_{i_o i_1}, \, a_{i_1 i_2},\,\dots,a_{i_\ell j}
\,\neq\, 0\,,$$
for suitable~$i_1,\dots,i_\ell$. Then, by~\eqref{cista}
and~\eqref{step}, we have that~$\Phi^\tau(u)_{i_1}=
\Phi^\tau(v)_{i_1}$ for any~$\tau\in[0,t_o]$.
Then, by applying~\eqref{step}
again, we get that~$\Phi^\tau(u)_{i_2}=
\Phi^t(v)_{i_2}$ 
for any~$\tau\in[0,t_o]$.
By repeatedly
applying~\eqref{step}, we thus conclude 
that~$\Phi^\tau(u)_{i_k}=\Phi^\tau(v)_{i_k}$ for~$k=1,\dots,\ell$ and 
so~$\Phi^\tau(u)_j=\Phi^\tau(v)_j$ for any~$\tau\in[0,t_o]$.
In particular,~$u_j=v_j$.

The proof of Lemma~\ref{strong:co}
is thus concluded.\hfill $\square$
\medskip

As a straightforward consequence of
Lemma~\ref{strong:co}, we obtain:

\begin{corollary}\label{strong:2}
Let the interaction be transitive.
Let~$u_i$ and~$v_i$ be two critical points in such a way
that~$u_i\ge v_i$ for any~$i\in\integer^d$ and~$u_{i_o}
> v_{i_o}$ for 
some~$i_o\in\integer^d$.

Then,~$u_i> v_i$ for any~$i\in\integer^d$.
\end{corollary}

Therefore, when the interaction is transitive, we 
see that when we consider a lamination of critical 
points, the sheets are disjoint, in more agreement 
with the common usage in topology. 
If the interaction is not transitive,
the conclusions of  Corollary~\ref{strong:2}
could be false. For example, if we consider a
layered material in which different layers do not interact, it is
easy to have minimizers so that they agree on one layer 
but are different -- but comparable -- on the others.



\section*{Proof of Theorem~\ref{minimizersexist}}

The proof we present
uses the rearrangement
method of~[CLl00] and it is based the construction
of a ``minimal minimizer'' which possesses additional
geometric properties.
Further details and generalizations are given in~[LlV06a].

We will first consider 
the case~$\sigma\in\rational^d$
and prove good uniform estimates. 
The case~$\sigma\in\real^d\setminus\rational^d$
is then obtained by a limit process. 

We fix some finite index
subgroup~${\mathcal{K}}$
of
\begin{equation}\label{resmod}
{\mathcal{K}}(\sigma):=
\{k\in\integer^d{\mbox{ s.t. }} \sigma\cdot k\in\integer\}
\end{equation}
and
we consider the
{\bf periodic configurations}
in
\begin{equation} \label{manif}
{\mathcal C}_{\mathcal{K}}
:=\Big\{ u{\mbox { s.t. }} \sup_{
i\in\integer^d}|u_i-\sigma\cdot i|\leq 10
{\mbox{ and }}
u_{i+j}=u_{i}+\sigma\cdot j
{\mbox{ for any }} j\in
{\mathcal{K}}
\Big\}\,.\end{equation}
In what follows, we identify, with a slight abuse
of notation, the quotient~$\integer^d/{{\mathcal{K}}}$
with an integer polytope
centered at the origin.

We can assume without loss of generality that the sides are much larger 
than the range of the interaction.  

We now {\bf quotient our variational problem} 
with respect to~${\mathcal{K}}$.
That is, given~$i\in\rational^d$,
we consider the equivalence class
$$ [i]:= \{ i'\in\rational^d {\mbox{ s.t. }}
i-i'\in {\mathcal{K}}
\}\,.$$
Analogously,
given~$i$, $j\in\rational^d$, we set
$$ [(i,j)]:= \{ (i',j')\in\rational^d\times\rational^d {\mbox{ s.t. }}
i-i'=j-j'\in{\mathcal{K}}
\}\,.$$
Given any~$i\in\rational^d$ (resp.,~$i$, $j\in\rational^d$),
we 
then choose one element~$i_{\mathcal{K}} \in [i]$
(resp., $(i_{\mathcal{K}},j_{\mathcal{K}})\in [(i,j)]$).

We also consider the quotient energy functional
\begin{equation}\label{def:s:ri}
S_{\mathcal{K}}(u):= 
\sum_{(i_{\mathcal{K}},
j_{\mathcal{K}})} a_{i_{\mathcal{K}}j_{\mathcal{K}}}
|u_{i_{\mathcal{K}}}-
u_{j_{\mathcal{K}}}-c_{ij}|^{p_{ij}} +\sum_{
i_{\mathcal{K}}}V(u_{i_{\mathcal{K}}})\,.
\end{equation}
The functional~$ S_{\mathcal{K}}$ is well defined for~$u\in
{\mathcal C}_{\mathcal{K}}$ and it does not depend on
the choice of~$i_{\mathcal{K}}$, $j_{\mathcal{K}}$.
We also note that the series in~\eqref{def:s:ri}
are just finite sums.
Indeed, the set~$\integer^d/{\mathcal{K}}$
is finite, since~$\sigma\in\rational^d$ and~${\mathcal{K}}$
has finite index in~${\mathcal{K}}(\sigma)$, thus we may
always think that the contribution of the above series
is given by some~$i_{\mathcal{K}}$ in such a finite set
and by~$j_{\mathcal{K}}$ satisfying~$|i_{\mathcal{K}}
-j_{\mathcal{K}}|<R$,
which, again, is a finite set.
Note that
the value of~$S_{\mathcal{K}}$ on~${\mathcal C}_{\mathcal{K}}
\ni u$
is determined once we know~$u_i$ for~$i\in
\integer^d/{\mathcal{K}}$, which is a finite set.
Therefore, minimizing~$S_{\mathcal{K}}$ 
on~${\mathcal C}_{\mathcal{K}}$
is a finite dimensional problem. 
It is easy to check that if the distance 
between two sites goes to infinity, the energy 
goes to infinity. The problem is also invariant 
by addition of an integer to the state.  
Consequently, 
the set~${\mathcal M}_{\mathcal{K}}$
of {\bf minimizers}
of ~$S_{\mathcal{K}}$ on~${\mathcal C}_{\mathcal{K}}$
is non empty.

Moreover, if~$u\in{\mathcal M}_{\mathcal{K}}$, then so 
does~$v:=u+k$ for any~$k\in
\integer$ and, if
$$k\geq \max_{i,\ell\in 
\integer^d/{\mathcal{K}} }|u_{i+\ell}|+|\sigma|
\max_{j\in 
\integer^d/{\mathcal{K}}} |j|\,,$$
we have that, given any~$i\in 
\integer^d/{\mathcal{K}}$ and~$j\in \integer^d$,
with~$j=j_1+j_2$, $j_1\in \integer^d/{\mathcal{K}}$
and~$j_2\in{\mathcal{K}}$,
$$ v_{i+j}=u_{i+j_1+j_2}+k=
u_{i+j_1}+\sigma\cdot j_2+k=
u_{i+j_1}+\sigma\cdot j-\sigma\cdot j_1+k
\ge \sigma\cdot j\,.$$
Consequently the set
$${\mathcal M}_{\mathcal{K}}^\star
:={\mathcal M}_{\mathcal{K}}
\cap
\{
u_{i+j}\ge\sigma\cdot j {\mbox { for any }}
i\in 
\integer^d/{\mathcal{K}}{\mbox { and }}
j\in \integer^d
\}$$
is non empty.

Given two configurations~$u_i$ and~$v_i$ we now
consider the configurations
$${\mbox{$\min\{u,v\}_i:=\min
\{u_i,v_i\}\quad$
and $\quad\max\{u,v\}_i:=\max
\{u_i,v_i\}$.}}$$
The following inequality will be fundamental for 
our argument:
\begin{equation}\label{eqder2} 
S_{\mathcal{K}}(\min\{u,v\})+
S_{\mathcal{K}}(\max\{u,v\})\leq
S_{\mathcal{K}}(u)+S_{\mathcal{K}}(v)\,.
\end{equation}
To prove~\eqref{eqder2},
we set~$\alpha_i:=\max\{u,v\}_i -u_i$
and~$\beta_i:=u_i-\min\{u,v\}_i$. Then~$\alpha_i \beta_i
\ge0$,
$\alpha_i\beta_i=0$ and $v_i=u_i+\alpha_i-\beta_i$. 
Thus, if we 
consider the
function
$$ [-1,1]^2\ni (s,t)\,\longmapsto\,
F(s,t):=|(u+t\alpha+s\beta)_i-
(u+t\alpha+s\beta)_j-c_{ij}
|^{p_{ij}}\,,$$
we have that
$$ \partial^2_{st} F=-\sum_{i,j} p_{ij}(p_{ij}-1)
|(u+t\alpha+s\beta)_i-
(u+t\alpha+s\beta)_j-c_{ij}
|^{p_{ij}-2} (\alpha_i\beta_j
+\alpha_j\beta_i)\le 0\,.$$
Accordingly,
\begin{eqnarray*}
0 &\geq&
\int_0^1 \int_{-1}^0
\partial^2_{st}F\,ds\,dt \\
&=&
|\max\{u,v\}_i-\max\{u,v\}_j -c_{ij}|^{p_{ij}}
+|\min\{u,v\}_i-\min\{u,v\}_j -c_{ij}|^{p_{ij}}\\ &&
-|u_i-u_j -c_{ij}|^{p_{ij}}
-|v_i-v_j -c_{ij}|^{p_{ij}}
\,,\end{eqnarray*} from which~\eqref{eqder2}
plainly follows.

The above considerations imply that
\begin{equation}\label{mimi} u^{\mathcal{K}}_i 
:=\min_{u\in{\mathcal M}_{\mathcal{K}}^\star} 
u_i\end{equation}
is also in~${\mathcal M}_{\mathcal{K}}^\star$ and it thus will
be called the {\bf minimal minimizer}.

One of the main feature of~$u^{\mathcal{K}}$ is that it
satisfies the {\bf no symmetry breaking} condition, namely,
that
\begin{equation}\label{no-s-b}
u^{\mathcal{K}}=u^{{\mathcal{K}}( \sigma)}\end{equation} 
for any finite index 
subgroup~${\mathcal{K}}$ of~${\mathcal{K}}
(\sigma)$.

To confirm~\eqref{no-s-b}, we argue as follows.
Given~$k\in{\mathcal{K}}(\sigma)$,
we observe that~$\tilde u\in{\mathcal M}_{\mathcal{K}}^\star$,
where~$\tilde u_i:=u^{{\mathcal{K}}}_{i+k}-\sigma\cdot k$, and 
therefore~$u^{{\mathcal{K}}}_i
\leq \tilde u_i$ and so
$$ u^{{\mathcal{K}}}_i \leq \min_{k\in{\mathcal{K}}(\sigma)} 
u^{{\mathcal{K}}}_{i+k}-\sigma\cdot k\,.$$
Since the other inequality is obvious,
we have that
$$ u^{{\mathcal{K}}}_i = \min_{k\in{\mathcal{K}}(\sigma)} 
u^{{\mathcal{K}}}_{i+k}-\sigma\cdot k$$
and so
that
$$ u^{{\mathcal{K}}}_i =  
u^{{\mathcal{K}}}_{i+k}-\sigma\cdot k$$
for any~${k\in{\mathcal{K}}(\sigma)}$.

In particular,~$ u^{{\mathcal{K}}}$ belongs 
to~${\mathcal C}_{{\mathcal{K}}(\sigma)}$ (and not only
to~${\mathcal C}_{{{\mathcal{K}}}}$).
This and the fact that
$$ S_{{\mathcal{K}}}
= \sharp ({{\mathcal{K}}(\sigma)}/{{\mathcal{K}}})\,
S_{{\mathcal{K}}(\sigma)}$$
on~${\mathcal C}_{{{\mathcal{K}}}(\sigma)}$
yield~\eqref{no-s-b}.

Also, if~$j$ and~$s$ are as in~\eqref{BP},
we have that~$v_i:= 
u^{{\mathcal{K}}( \sigma)}_{i+j}-s\in
{\mathcal M}_{\mathcal{K}}^\star$ and therefore~$v_i\ge
u^{{\mathcal{K}}( \sigma)}_i$, yielding the
{\bf order property} in~\eqref{BP}.

Let now~$j\in \integer^d$. Take~$s$ such that~$\sigma\cdot j\in
[s,s+1]$. Then, the order property in~\eqref{BP}
implies that
$$ \sigma\cdot j-1\leq s\leq 
u^{{\mathcal{K}}( \sigma)}_{i+j}-
u^{{\mathcal{K}}( \sigma)}_i\leq s+1\leq \sigma\cdot j+1$$
thence
\begin{equation}\label{sti:l}
\sup_{i,j\in\integer^d}
|u^{{\mathcal{K}}( \sigma)}_{i+j}-
u^{{\mathcal{K}}( \sigma)}_i - \sigma\cdot j|\le 1\,.
\end{equation}
We also observe that there must be
some~$j_o\in \integer^d$ such that~$
u^{{\mathcal{K}}( \sigma)}_{j_o}-\sigma\cdot j_o
\in [0,1]$. Indeed, if not, the fact that~$
u^{{\mathcal{K}}( \sigma)}\in {\mathcal M}_{\mathcal{K}}^\star$
would give that~$u^{{\mathcal{K}}( \sigma)}_{j}-\sigma\cdot j\geq 1$
for any~$j\in\integer^d$, and so~$v_i:=u^{{\mathcal{K}}( \sigma)}_{i}
-1$ would also lie in~$ {\mathcal M}_{\mathcal{K}}^\star$,
in contradiction with the definition of minimal minimizer
given in~\eqref{mimi}.

Then, by~\eqref{sti:l},
$$
|u^{{\mathcal{K}}( \sigma)}_{i}-\sigma\cdot i|\leq
|u^{{\mathcal{K}}( \sigma)}_{i}-u^{{\mathcal{K}}( \sigma)}_{j_o}
-\sigma\cdot (i-j_o)|+1\leq 2
\,,$$
for any~$i\in\integer^d$,
thus giving~\eqref{BondUn}
and the fact that~$u^{{\mathcal{K}}( \sigma)}$
does not touch the boundary of~${\mathcal{C}}_{\mathcal{K}}$ as defined
in~\eqref{manif}.

Also,~$u^{{\mathcal{K}}( \sigma)}$
{\bf is a ground state}. Indeed,
let~$K>0$ and~$v_i$ be a configuration such that~$v_i=
u^{{\mathcal{K}}( \sigma)}_i$ for~$|i|>K$.
We take~$\ell\in \nat$ so large that
the set
$$ \{ j\in\integer^d {\mbox{ s.t. }}
|i-j|\leq R {\mbox{ for some }} |i|\leq K\}
$$
is contained in
the fundamental polytope~$\integer^d/(\ell{{\mathcal{K}}( \sigma)})$.

We now introduce the ``periodic extension'' of~$v$.
Namely, given~$i\in \integer^d$ with~$i=i_1+i_2$,
$i_1\in \integer^d/(\ell{\mathcal{K}}(\sigma))$
and~$i_2\in\ell{\mathcal{K}}(\sigma)$,
we define~$w_i:= v_{i_1}+\sigma\cdot i_2$. 
Then,~$w\in {\mathcal C}_{\ell\mathcal{K}(\sigma)}$
and so
$$ S_{\ell\mathcal{K}(\sigma)} (w) \geq
S_{\ell\mathcal{K}(\sigma)} (u^{\ell{\mathcal{K}}( \sigma)})=
S_{\ell\mathcal{K}(\sigma)} (u^{{\mathcal{K}}( \sigma)})
\,.$$
This says that
\begin{eqnarray*}
&&\sum_{i\in \integer^d/(\ell{\mathcal{K}}(\sigma)),
\,|j-i|\leq R} a_{ij}
|w_i-
w_j-c_{ij}|^{p_{ij}} +\sum_{
i\in\integer^d/(\ell{\mathcal{K}}(\sigma))}V(w_i)
\\
&\geq&\sum_{i\in \integer^d/(\ell{\mathcal{K}}(\sigma)),
\,|j-i|\leq R} a_{ij}
|u^{{\mathcal{K}}( \sigma)}_i-
u^{{\mathcal{K}}( \sigma)}_j-c_{ij}|^{p+{ij}} +\sum_{
i\in\integer^d/(\ell{\mathcal{K}}(\sigma))
}V(u^{{\mathcal{K}}( \sigma)}_i)
\,.\end{eqnarray*}
Note that
$$ \sum_{
i\in\integer^d/(\ell{\mathcal{K}}(\sigma))
}V(w_i)
=
\sum_{|i|\le K}V(v_i)
+
\sum_{{
i\in\integer^d/(\ell{\mathcal{K}}(\sigma))
}\atop{|i|>K}}V(u^{{\mathcal{K}}( \sigma)}_i)
\,.$$
Also, it is easily seen that if~$i\in\integer^d/{{\mathcal{K}}( \sigma)}$,
$|i-j|\le R$ and~$|j|>K$, we have that~$|j+h|>K$
for any~$h\in
{{\mathcal{K}}( \sigma)}$ and so
\begin{eqnarray*}
&& \sum_{i\in \integer^d/(\ell{\mathcal{K}}(\sigma)),
\,|j-i|\leq R} a_{ij}
|w_i-
w_j-c_{ij}|^{p_{ij}}\\&=&
\sum_{|i|\le K}a_{ij}
|v_i-
v_j-c|^{p_{ij}}
+\sum_{{i\in \integer^d/(\ell{\mathcal{K}}(\sigma)),
\,|j-i|\leq R}\atop{|i|>K\ge |j|}} a_{ij}
|v_i-v_j-c_{ij}|^{p_{ij}}
\\&&+   
\sum_{{i\in \integer^d/(\ell{\mathcal{K}}(\sigma)),
\,|j-i|\leq R}\atop{|i|,|j|>K}} a_{ij}
|v_i-
w_j-c_{ij}|^{p_{ij}}\\ &=&
\sum_{
\min\{|i|,|j|\}\le K}
a_{ij}
|v_i- 
v_j-c_{ij}|^{p_{ij}}
+
\sum_{{i\in \integer^d/(\ell{\mathcal{K}}(\sigma))}\atop{
|i|,|j|>K
}} a_{ij}
|u^{{\mathcal{K}}( \sigma)}_i-
u^{{\mathcal{K}}( \sigma)}_j-c_{ij}|^{p_{ij}}
\,.
\end{eqnarray*}
The above considerations yields
that~$u^{{\mathcal{K}}( \sigma)}$
is a ground state.

The lamination property is a consequence of~\eqref{BP}.

This ends the proof of
Theorem~\ref{minimizersexist}.\hfill $\square$

\section*{Proof of Theorem~\ref{MAIN:TH}}

The proof we provide is based on the gradient flow method
of~[A90,KLlR97, CLl98, LlV05].
Further details and generalizations are given in~[LlV06b].

We fix~$\sigma\in\real^d$ (not necessarily rational) and
we consider the ``resonant set''~${\mathcal{K}}(\sigma)$ 
given 
in~\eqref{resmod}.
For~$\ell\in \nat$, with a slight abuse of notation, we
identify
the quotient spaces~$\integer^d
/(\ell{\mathcal{K}}(\sigma))$
with a sequence of nested fundamental
domains, that is, each~$\integer^d
/(\ell{\mathcal{K}}(\sigma))$ is a polytope
centered at the origin. Note that, when~$\sigma\in
\real^d\setminus\rational^d$, some sides of such
a polytope
may have infinite length
and, in fact, if~$
{\mathcal{K}}(\sigma)=
\{0\}$ then~$\integer^d
/(\ell{\mathcal{K}}(\sigma))=\integer^d$.
In any case,~$\integer^d
/(\ell{\mathcal{K}}(\sigma))\subset
\integer^d
/((\ell+1){\mathcal{K}}(\sigma))\subset\dots$
is a sequence of polytopes
invading the whole~$\integer^d
$.
We also remark that,
as a consequence of the periodicity and
lamination properties,
\begin{equation}\label{wrap on the T}
\sum_{i\in \integer^d
/(\ell{\mathcal{K}}(\sigma))} |u^{(1)}_i-u^{(0)}_i|\le
\sharp\Big({\mathcal{K}}(\sigma)/
(\ell{\mathcal{K}}(\sigma))
\Big)\,.\end{equation}
We define
the space
$$ {\mathcal{C}}:=\{{\mbox{$u_i$ satisfying 
\eqref{BP} s.t. $u^{(0)}_i\leq u\leq u^{(1)}_i$
for any $i\in \nat$}}\}$$
and, for~$u$, $v\in{\mathcal{C}}$,
we consider the {\bf renormalized energy}
\begin{eqnarray}\label{reeen}
S_\ell(u,v) &:=&\nonumber
\sum_{ i\in 
\integer^d
/(\ell{\mathcal{K}}(\sigma)),
j\in\integer^d } a_{i j}\Big(
|u_i-
u_j-c_{ij}|^{p_{ij}} 
-|v_i-
v_j-c_{ij}|^{p_{ij} } 
\Big)\\&&
+\sum_{
i\in \integer^d
/(\ell{\mathcal{K}}(\sigma))} 
\Big(V(u_i)-V(v_i)\Big)\,.
\end{eqnarray}
Note that
\begin{equation}\label{12bis}
\begin{split} & u_i-u_j\pm c_{ij}\le u_i^{(1)}-u_j^{(0)}+\Lambda
\\ &\quad\le |u_i^{(1)}-\sigma
\cdot i|+|\sigma\cdot j-u_j^{(0)}|+|\sigma|\,|i-j|+\Lambda
\le 4+|\sigma|\,R+\Lambda\,,\end{split} \end{equation}
if~$u\in{\mathcal{C}}$
and~$|i-j|\le R$, because of~\eqref{BondUn}.
Then, the series in~\eqref{reeen}
are convergent,
thanks to~\eqref{wrap on the T}
and~\eqref{12bis}.

Notice also that if~$v^{(n)}\in{\mathcal{C}}$ is so
that
$$ \lim_{n\rightarrow+\infty} v^{(n)}_i=v_i
\;\,{\mbox{ for any }}\,i\in \integer^d/(\ell {\mathcal{K}}(\sigma))$$
it follows from~\eqref{wrap on the T}
and the Dominated
Convergence Theorem that
$$\lim_{n\rightarrow+\infty} 
\sum_{i\in
\integer^d/(\ell {\mathcal{K}}(\sigma))
}
|v^{(n)}_i-v_i|\,=\,0\,,$$
thence the pointwise and uniform convergence
agree in~${\mathcal{C}}$.

We now show that {\bf
the renormalized energy
is good in detecting the ground states}.
Namely, we prove that, if~$u\in
{\mathcal{C}}$,
\begin{equation}
\label{en:equi}
{\mbox{$S_\ell(u, u^{(0)}) \ge 0$ and equality holds
if and only if 
$u$ is a ground state.}}
\end{equation}
The proof of such a statement will be accomplished
in several steps.
First, we show that
\begin{equation}\label{mag0}
{\mbox{if
$v$ is a ground state, 
$S_\ell(u,v)\ge 0$.}}
\end{equation}
In the proof of~\eqref{mag0},
we may assume, without loss of generality, that
\begin{equation}\label{mag1}
{\mbox{$u_i
=v_i$ if~$|\sigma\cdot i|>A|\sigma|$, for some~$A>0$.
}}\end{equation}
Indeed, once~\eqref{mag0} is proven under such additional
assumption, we define
$$ u^\star_i\,:=\,\left\{
\begin{matrix}
u_i & &{\mbox{ if $|\sigma\cdot i|\le A|\sigma|$,}}\\
v_i & &{\mbox{ if $|\sigma\cdot i|> A|\sigma|$,}}
\end{matrix}\right.
$$
and we use~\eqref{12bis} and~\eqref{wrap on the T}
to obtain~\eqref{mag0} for~$u$ from the same
result for~$u^\star$, by sending~$A\rightarrow+\infty$.

Then, we focus on the proof of~\eqref{mag0}
under the additional assumption~\eqref{mag1}.
To this extent, 
we denote here by~$C$ suitable positive constants
which may depend on~$\sigma$, $A$, $\ell$ and on the structural
constants of the problem, and which may be different at
different steps of the computation.  
We let~$r:={\, \rm dim\,}  ({\mathcal{K}} (\sigma) )$.
We suppose~$r\ge 1$ (if~$r=0$,~\eqref{mag0}
follows by a cut-off argument and~\eqref{wrap on the T}).
We also take~$n$ and~$K$ to be large integers in such
a way that~$K\in [n\ell-C, \,n\ell+C]$ and~$\{ j\in \integer^d 
{\mbox{ s.t. }} |i-j|\le R {\mbox{ for some }} i\in \integer^d,\;\,
|i|\le K \}\subseteq \integer^d/(n\ell
{\mathcal{K}} (\sigma) )$.
We define
$$ w_i\,:=\,\left\{
\begin{matrix}
u_i & &{\mbox{ if $|i|< K$,}}\\
v_i & &{\mbox{ if $|i|\ge K$}}
\end{matrix}\right.
$$
and we deduce from the ground state
property
of~$v$ that
$$ \sum_{\min\{ |i|,|j|\} \le K}
a_{ij}\Big( 
|w_i-w_j-c_{ij}|^{p_{ij}}
-|v_i-v_j-c_{ij}|^{p_{ij}}
\Big)+\sum_{|i|\le K}\Big( V(w_i)-V(v_i)
\Big)
\ge 0
\,.$$
We use the latter inequality and~\eqref{12bis}
to gather that
\begin{eqnarray*}
\sharp \left( \frac{\ell {\mathcal{K}} (\sigma)}{
n\ell {\mathcal{K}} (\sigma)}\right)\, S_\ell(u,v)&=&
S_{n\ell}(u,v) \\
&\geq& -C\,\sum_{
{j\in
\integer^d/(n\ell{\mathcal{K}} (\sigma) )}\atop{
{{|\sigma\cdot j|\le C|\sigma|}\atop{|j|\ge K-C}}}
}|u_j-v_j|\\
&\geq& -C\,\sharp \Big\{
j\in 
\integer^d/(n\ell{\mathcal{K}} (\sigma) )\,,\;\,
|\sigma\cdot j|\le C|\sigma|\,,\,\;|j|\ge K-C
\Big\}
\\ &\ge& -C\,(n\ell)^{r-1}\,.
\end{eqnarray*}
We then divide both sides by~$ \sharp 
\left( \frac{\ell {\mathcal{K}} (\sigma)}{
n\ell {\mathcal{K}} (\sigma)}\right)\sim (n\ell)^r$
and we take~$n$ arbitrarily large in order
to obtain
that~$S_\ell(u,v)\ge 0$.
This ends the proof of~\eqref{mag0}.

By exchanging the r\^oles of~$u$ and~$v$,
we also deduce from~\eqref{mag0}
that
\begin{equation}\label{ug0}
{\mbox{if
$u$ and $v$ are ground states, 
$S_\ell(u,v)= 0$.}}\end{equation}
We now show that
\begin{equation}\label{gs0}
{\mbox{if
$v$ is a ground state and
$S_\ell(u,v)= 0$ then
$u$ is a ground state too.}}\end{equation}
To prove this, let~$w$ be such that~$w_i=u_i$ if~$|i|>K$.
Let~$n_o$ so large that
\begin{equation}\label{conta}
\{ j\in \integer^d 
{\mbox{ s.t. }} |i-j|\le R {\mbox{ for some }}  i\in \integer^d,
|i|\le K \}\subseteq \integer^d/(n_o \ell
{\mathcal{K}} (\sigma) )\,.
\end{equation}
We then use
the fact that~$S_\ell (u,v)=0$
and~\eqref{mag0}
to conclude that
$$
S_{n_o\ell}(u,w)=
S_{n_o\ell}(u,v)+
S_{n_o\ell}(v,w)=
0-S_{n_o\ell}(w,v)\le 0\,,
$$
which implies that~$u$ is
a ground state, thanks to~\eqref{conta}.

By collecting the results of~\eqref{mag0},
\eqref{ug0}
and~\eqref{gs0}, the claim in~\eqref{en:equi}
follows.

We also introduce the following~{\bf gradient flow}.
Given~$u_i\in{\mathcal{C}}$, we consider the configuration~$\Phi^t(u)_i$,
with~$i\in \nat$ and~$t\ge 0$ to be a solution of the Cauchy
Problem~\eqref{Cauc}, which turns out to be
the gradient flow
of~\eqref{reeen}.

The right hand side of~\eqref{Cauc} 
is locally Lipschitz in~$\ell^\infty$ 
so the flow satisfies existence and uniqueness for short times. 
The Comparison Principle in Lemma~\ref{weak:co}
implies 
that if~$ u_i \ge v_i$
for all~$i \in \integer^d$, then~$\Phi^t (u)_i \ge \Phi^t(v)_i$
for all~$i \in \integer^d$.
Hence we conclude that~$\Phi^t({\mathcal{C}})\subseteq{\mathcal{C}}$.

Moreover, since, for  functions in~$\mathcal{C}$ 
the generator of the flow is uniformly Lipschitz, we conclude 
that~$\Phi^t$ extends for all time for
initial data in~$\mathcal{C}$, due to the 
Prolongation Lemma (see~[H80]).

Furthermore, 
\begin{equation}\label{bulj}
-\partial_t S_\ell (\Phi^t(u),u^{(0)})
= \sum_{i\in \integer^d/(
\ell{\mathcal{K}}(\sigma)
)} \Big( \partial_t \Phi^t(u)_i\Big)^2
\ge 0\,.\end{equation}
Note that the vanishing of the quantity in~\eqref{bulj}
is the same as being a  critical point in~\eqref{CP}.

We now look at the
{\bf evolution of the linear interpolation} between~$u^{(0)}$
and~$u^{(1)}$, that is, for~$s\in[0,1]$, we define
$$ u^{(s)}:= s u^{(1)}+(1-s) u^{(0)}$$
and we conclude by the above construction and
Tychonoff Theorem that for any~$s\in[0,1]$
there is a sequence of times~$t_n(s)$ 
and a critical point~$u^{(s,\star)}$
such that
$$
\lim_{n\rightarrow+\infty} t_n(s)=+\infty
\,{\mbox{ and }}\,
\lim_{n\rightarrow+\infty}\Phi^{t_n(s)}(u^{(s)})_i=
u^{(s,\star)}_i
\,.$$
The proof of
Theorem~\ref{MAIN:TH}
will be completed once we are able to show that there 
exists~$s\in [0,1]$
in such a way that~$
u^{(s,\star)}$ does not coincide with~$
u^{(0)}$ nor with~$
u^{(1)}$.

For this, given~$r\ge 0$,
we consider the {\bf energy ball}
$$B_r^\star:=
\{
u\in {\mathcal{C}} {\mbox{ s.t. }} 
S_\ell(u,u^{(0)})\le r
\}\,.$$
Then, in the light of~\eqref{en:equi},
we may suppose that
\begin{equation} \label{bolg}
{\mathcal{C}} \cap B^\star_0 =
\{u^{(0)},u^{(1)}\}\,,\end{equation} 
otherwise
another ground state (and so, a fortiori a critical point) would
exist, thus ending the proof. 

We now argue {\bf by contradiction} and assume
that no other critical point but~$u^{(0)}$ and~$u^{(1)}$
lies in~${\mathcal{C}}$.
It thus follows from~\eqref{en:equi}
and~\eqref{bolg} that, for small~$r>0$,
the energy ball~$B_r^\star$ is disconnected, i.e.,
it has a connected component~$C^{(0)}$ near~$u^{(0)}$ and
another connected component~$C^{(1)}$
near~$u^{(1)}$. This gives that once~$\Phi^t(u^{(s)})$
has entered~$C^{(0)}$ (resp.,~$C^{(1)}$) it cannot leave it
and it thus approaches~$u^{(0)}$ (resp.,~$u^{(1)}$).

This considerations allow us to define the
{\bf basin of attraction}~$\beta^{(0)}$ (resp.,~$\beta^{(1)}$)
as all the~$s\in[0,1]$ in such a way~$\Phi^t(u^{(s)})$
approaches~$u^{(0)}$ (resp.,~$u^{(1)}$) as~$t\rightarrow+\infty$.
The continuous dependence of~$\Phi^t$ 
on initial data implies that~$\beta^{(0)}$
and~$\beta^{(1)}$ are open.
But then, ~$\beta^{(0)}$ and~$\beta^{(1)}$
are open and disjoint and they cover~$[0,1]$,
which is  a contradiction.

This completes the proof of
Theorem~\ref{MAIN:TH}.\hfill $\square$

\section*{Proof of Corollary~\ref{MAIN:CO}}

The desired result plainly follows from
Theorem~\ref{MAIN:TH} and
Corollary~\ref{strong:2}.\hfill $\square$

\bigskip

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\bigskip

The preprints~[LlV05], [LlV06a]
and~[LlV06b]
are available on-line at \begin{center} {\tt 
http://www.ma.utexas.edu/mp\_arc/
}\end{center}

\bigskip\bigskip


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