\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 15, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/15\hfil Existence of lattice solutions]
{Existence of lattice solutions to semilinear elliptic systems
with periodic potential}

\author[N. D. Alikakos, P. Smyrnelis \hfil EJDE-2012/15\hfilneg]
{Nicholas D. Alikakos, Panayotis Smyrnelis} % in alphabetical order

\address{Nicholas D. Alikakos \newline
Department of Mathematics, University of Athens,
Panepistemiopolis, 15784 Athens, Greece \newline
Institute for Applied and Computational Mathematics\\
Foundation of Research and Technology - Hellas\\
71110 Heraklion, Crete, Greece}
\email{nalikako@math.uoa.gr}

\address{Panayotis Smyrnelis \newline
Department of Mathematics\\
Aristotle University of Thessaloniki\\
54124 Thessaloniki, Greece}
\email{ysmyrnelis@yahoo.fr}

\thanks{Submitted October 24, 2011. Published January 24, 2012.}
\thanks{N. D. A. was supported by the Kapodistrias grant
15/4/5622 from the University of Athens}
\subjclass[2000]{35J20, 35J50}
\keywords{Lattice solution; invariant; periodic Potential; elliptic system; 
\hfill\break\indent reflection; discrete group; variational calculus}

\begin{abstract}
 Under the assumption that the potential $W$ is invariant
 under a general discrete reflection group $G'=TG$ acting
 on $\mathbb{R}^n$, we establish existence of $G'$-equivariant
 solutions  to $\Delta u - W_u(u) = 0$, and find an estimate.
 By taking the size of the cell of the lattice in space domain
 to infinity, we obtain that these solutions converge to
 $G$-equivariant solutions connecting the minima of the potential
 $W$ along certain directions at infinity. When particularized to
 the nonlinear harmonic oscillator $u''+\alpha \sin u=0$, $\alpha>0$,
 the solutions  correspond to those in the phase plane
 above and below the heteroclinic connections,
 while the $G$-equivariant solutions captured in the limit
 correspond to  the heteroclinic connections themselves.
 Our main tool is the $G'$-positivity of the parabolic semigroup
 associated with the  elliptic system which requires only
 the hypothesis of symmetry  for $W$. The constructed solutions
 are positive in the sense that  as maps from $\mathbb{R}^n$ into
 itself leave the closure of the fundamental alcove (region) invariant.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks

\section{Introduction}

The study of the system
\begin{equation}\label{system}
 \Delta u - W_u(u) = 0, \quad \text{for } u:
 \mathbb{R}^n \to \mathbb{R}^n,
\end{equation}
where $W: \mathbb{R}^n \to \mathbb{R}$ and
$W_u := (\partial W / \partial u_1, \dots, \partial W /
\partial u_n)^T$ under symmetry hypotheses on the potential
$W$, was initiated in Bronsard, Gui, and Schatzman
\cite{bronsard-gui-schatzman}, where existence for the case
$n=2$ with the symmetries of the equilateral triangle was settled,
and followed later by Gui and Schatzman \cite{gui-schatzman},
where the case $n=3$ for the symmetry group of the tetrahedron
was established. The corresponding solutions are known as the
\emph{triple junction} and the \emph{quadruple junction} respectively.
This class of solutions is characterized by the fact that they
\emph{connect} the global minima of the potential $W$ in certain
directions as $|x|\to+\infty$. In \cite{alikakos-fusco},
Alikakos and Fusco established an abstract theorem for the
existence of such solutions, together with an estimate,
for general dimension $n$ and \emph{any reflection point group $G$}
acting on $\mathbb{R}^n$. Finally in \cite{alikakos} %insert reference
one of us gave a simpler proof of the result in \cite{alikakos-fusco}.

Reflections are special linear isometries, and as such leave the
Laplacian invariant. The groups of symmetries of the equilateral
triangle and the tetrahedron are reflection groups.
This motivated the study in \cite{alikakos-fusco} for general
reflection groups. By including also translations, which obviously
leave the Laplacian invariant, we are led naturally to
\emph{discrete reflection groups} $G'$ acting on $\mathbb{R}^n$.
We note that $G'=TG$ where $T$ is the translation group of $G'$
and $G$ the point group fixing the origin $O$.
$T$ generates a \emph{lattice} in $\mathbb{R}^n$ via the orbit
$\{\tau O : \tau \in T\}$. There are two natural problems that
one can identify in this context:
\begin{itemize}
\item[(I)] The periodic potential problem;

\item[(II)] The periodic solution problem.
\end{itemize}

In Problem (I), we require that $W(gu)=W(u)$, for every
$u \in \mathbb{R}^n$ and every $g\in  G'$. Both domain and
target have $G'$ acting on them and the solutions are
$G'$-equivariant. Actually it is more appropriate to scale $G'$
in domain and target differently by introducing a parameter $R>0$:
\begin{equation}\label{G'_R}
G'_{R}:=\{ g_{R} : g_{R}(x):=R g(x/R) ,  g \in G' \}
\end{equation}
and then seek $(G'_{R},G')$ \textit{equivariant} solutions:
\begin{equation}\label{G'RG'}
u(g_{R}x)=gu(x) \quad \text{for every $x \in \mathbb{R}^n$  and every }
g \in G'.
\end{equation}


In Problem (II), $G'_{R}$ is acting in the domain and $G$ in the
target. We require that $W(gu)=W(u)$, for every
$u \in \mathbb{R}^n$ and $g\in G$, and we seek a solution in the
class of $G$-equivariant maps modulo the translations of the
discrete group $G'_{R}$:
\begin{equation}\label{G'_R G}
u(\tau_{R}gx)=gu(x) \quad \text{for every $x \in \mathbb{R}^n$,
 every $\tau \in T$,  and every $g \in G$}.
\end{equation}

These two classes of solutions are best visualized in terms of an
elementary example, the harmonic oscillator
$u''+\alpha \sin u =0$, $\alpha>0$, that we explain in detail below.
Type (I) correspond to the solutions above and below the separatrices
in the phase plane, while type (II) correspond to the periodic
solutions inside a single cell. The heteroclinic solutions
(separatrices) correspond to the solutions constructed
in \cite{bronsard-gui-schatzman,gui-schatzman,alikakos-fusco}.

In this paper we focus on problem (I).
We construct an equivariant solution to \eqref{system} which maps each
cell in the domain lattice to the corresponding cell in the target
lattice. Moreover, the map $u_{R}$ in the limit
$R\to\infty$, converges to another solution to \eqref{system}
equivariant for the point group $G$. In terms of the harmonic
oscillator above, the parameter $R$ relates to the ``time" $x$
that it takes for the solution $u_{R}$ to transverse a cell,
and thus as $R\to\infty$ the solution converges to the heteroclinic
connection\footnote{N. D. A. is indebted to Peter Bates for
pointing this out to him, as well as the different scalings.}.

One of the main tools in \cite{alikakos-fusco,alikakos},
is the positivity invariance of the gradient flow
$u_{t}=\Delta u - W_u(u)$, under appropriate boundary conditions.
A map $u$ is \textit{positive} if
$u ( \overline{F} ) \subset \overline{F}$, where $F$ is the
fundamental region of the group. Positivity is built in the
minimization process (that is, in constructing $u_R$) as a constraint.
This constraint is then removed via the gradient flow which is shown
to preserve it.
The solutions we construct are global minimizers in the class
of positive maps. It is not known if the property of positivity
is automatic for global minimizers.

In the point reflection group case treated
in \cite{alikakos-fusco} one can reduce the problem of positivity to
a half-space determined by a reflection plane (instead of $F$),
intersected by a ball of radius $R$, and homogeneous Neuman
conditions on the circumference. In the discrete reflection group
case treated in the present paper, the fundamental alcove is a
bounded simplex and one has to deal with the whole object all at once.

If $S_i$ denotes the ith face of the fundamental alcove and
$\Pi_i$ its supporting plane then the boundary conditions take
the form
\begin{equation}\label{boundaryconditions}
\begin{gathered}
x \in S_i \Rightarrow u(x)\in \Pi_i \\
x \in S_i \Rightarrow \frac{\partial u}{\partial r_{i}} (x)
\: \bot  \: \Pi_i ,\text{ where } r_{i} \: \bot \: \Pi_i \,.
\end{gathered}
\end{equation}

The main contribution in the present paper is the positivity
result under \eqref{boundaryconditions} (Theorem \ref{thm3}).
Otherwise the paper proceeds as in \cite{alikakos}.
We mention in passing that the boundary conditions in
\eqref{boundaryconditions} are conformal and for smooth domains
have been studied by Hamilton \cite{hamilton}.

\section{Notation}

As it was mentioned in the Introduction, $G'$ denotes a discrete
reflection group acting on $\mathbb{R}^n$, $G$ its associated
point group, and $G'_R$ the image of $G'$ by the homothetic
transformation $x \rightarrow Rx$, where $R>0$. $G'_R$ is the
scaled discrete group of the blown-up lattice; its elements are
defined in \eqref{G'_R}.

A discrete reflection group $G'$ is generated by a finite set
of reflections. The hyperplanes $\{x \in\mathbb{R}^n :  gx=x \}$
 associated to these reflections $g \in G'$, bound a region
called \textit{fundamental alcove} (see \cite{humphreys}) of $G'$
and denoted by $F'$, with the following properties:
\begin{itemize}
\item[(i)] $F'$ is an open polyhedron with acute angles
\footnote{Thanks to Christos Athanasiadis of the University of
Athens who confirmed this fact to us.},
\item[(ii)] $F' \cap gF' = \emptyset$, for $I \neq g \in G'$,
where $I$ is the identity,
\item[(iii)] $\mathbb{R}^n = \cup\{ g \overline{F'} : g \in G' \}$.
\end{itemize}

Similarly the associated point group $G$ has a
 \emph{fundamental region}, that is, a subset $F \subset \mathbb{R}^n$
with the following properties:
\begin{itemize}
\item[(i)] $F$ is open and convex,
\item[(ii)] $F \cap gF = \emptyset$, for $I \neq g \in G$, where $I$ is the identity,
\item[(iii)] $\mathbb{R}^n = \cup\{ g \overline{F} : g \in G \}$.
\end{itemize}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig1}
\end{center}
\caption{Fundamental alcoves for the discrete groups $G'_{R}$ and
$G'$ of the hexagonal lattice}
\label{fig1}
\end{figure}


$F'$ and $F$ can be chosen such that the origin $O$ is a vertex
of $F'$ and $F' \subset F$. As regards the scaled group $G'_R$,
it is clear that its fundamental alcove denoted by $F'_R$ is
the set $\{Rx : \ x \in F' \}$. We shall denote the closed faces
of $F'$ (respectively $F'_R$) by $S_i$, $i=1,\cdots,N$
(respectively $S^R_i$), and by $\Pi_i$ (respectively $\Pi^R_i$)
their supporting planes.

In the same way we defined in \eqref{G'RG'} $(G'_{R},G')$ equivariant
solutions, a map $u$ is called $G$ equivariant when:
\begin{equation}\label{G}
u(gx)=gu(x) \quad \text{for every $x \in \mathbb{R}^n $  and every }
 g \in G.
\end{equation}

Given $a \in \mathbb{R}^n$, the \emph{stabilizer} of $a$,
denoted by $G'_a$, is the subgroup of $G'$ that fixes $a$.
Finally, we shall denote by $\langle\cdot ,\cdot \rangle$
the Euclidean inner product, by $| \cdot |$ the Euclidean norm,
by $d( \cdot, \cdot)$ the Euclidean distance, and by $B(x,\epsilon)$
the open ball of center $x$ and radius $\epsilon$ with respect to
this distance.

\section{Main Theorems}

We begin by stating the  hypotheses.

\begin{itemize}
\item[(H1)] (Symmetry) %\label{h1}
The potential $W $, of class $C^3$, is invariant under a discrete
reflection group $G'$ acting on $\mathbb{R}^n$; that is,
\begin{equation}\label{g-invariance}
W(gu) = W(u) \quad \text{ for all $g \in G'$  and }u \in \mathbb{R}^n.
\end{equation}


\item[(H2)] (nondegenerate global minimum) %\label{h2}
Let $F' \subset \mathbb{R}^n$ be a fundamental alcove of $G'$.
We assume that $W$  is non-negative and has a unique zero $a$
in $\overline{F'}$. Furthermore, there holds
$v^\top\partial^2 W(u)v \geq 2 c^2 |v|^2$, for $v\in\mathbb{R}^n$
and $|u-a| \leq \bar{q}$, for some $c$, $\bar{q} > 0$.


\item[(H3)] ($Q$-monotonicity) % \label{h3}
We restrict ourselves to potentials $W$ for which there is a
continuous function $Q: \mathbb{R}^n \to \mathbb{R}$ that satisfies
\begin{equation}\label{q-smooth}
Q(u+a) = |u| + H(u),
\end{equation}
where $H:\mathbb{R}^n\to\mathbb{R}$ is a $C^2$ function such that
$H(0) = 0$ and $H_u(0) = 0$, and
\begin{subequations}\label{q-list}
\begin{gather}
Q \text{ is convex,}\label{q-list-a}\\
Q(u) >0,\quad \text{on } \mathbb{R}^n \setminus \{ a \}\label{q-list-d},
\end{gather}
\end{subequations}
and, moreover,
\begin{equation}\label{q-monotonicity}
\langle Q_u(u), W_u(u) \rangle \geq 0,\quad
\text{ in } D'\setminus \{ a \},
\end{equation}
where we have set
\begin{equation}
D':= \mathrm{Int}\left( {\cup_{g\in G'_{a}} g\overline{F'}} \right).
\end{equation}
\end{itemize}

\begin{theorem}\label{thm1}
Under Hypothesis {\rm (H1)},  for every $R>0$ there exists a
$(G'_{R},G')$ equivariant classical solution $u_R$ to
system \eqref{system} such that $u_R( F'_R)\subset F'$.
Furthermore, if {\rm (H2)--(H3)} also hold, then there
exist positive constants $R_0$, $k$, $K,$ such that for $R>R_0$
and $x \in D'_R:=\{Rx :  x \in D' \}$
\begin{equation}
|u_R(x)-a| \leq K \mathrm{e}^{-k d(x,\partial D'_R)} \,.
\end{equation}
\end{theorem}

\begin{theorem}\label{thm2}
Under Hypotheses {\rm (H1)--(H3)}, there exists a
$G$ equivariant classical solution $u$ to system \eqref{system}
such that
\begin{itemize}
\item[(i)] $u( \overline{F})\subset \overline{F'}$ and
$u(D)\subset D'$, where we have set
$D:= \cup_{R>0} \{Rx : x \in D' \}$.
\item[(ii)] $|u(x)-a| \leq K \mathrm{e}^{-k d(x,\partial D)}$,
for $x \in D$ and for positive constants $k$, $K$.
\end{itemize}
\end{theorem}

It would be interesting to relate Theorem \ref{thm2} to the main
result in \cite{alikakos-fusco}.
 Theorem \ref{thm1} was conjectured by G. Fusco in a personal
communication.

The harmonic oscillator is the elementary example which best
illustrates the above theorems. For the ODE,
\begin{equation}\label{oscillator eq}
u''+\alpha \sin u=0 ,\quad \text{with } \alpha>0,
\end{equation}
the corresponding potential is $W(u)= \alpha (1+ \cos u)$ which
is invariant under the discrete group $G'$ acting on $\mathbb{R}$,
generated by the reflections $s_0$ and $s_\pi$ with respect to
the points $0$ and $\pi$. The associated point group of $G'$
is $G= \{I, s_0 \}$, where $I$ denotes the identity.
The fundamental alcove of $G'$ is $F'=(0, \pi)$ while the
fundamental region of $G$ is $F=(0, \infty)$. Clearly,
$W$ has in $\overline{F'}$ a unique minimum attained at $a=\pi$,
and since $a \in \partial{F'}$, we have $D'=(0, 2 \pi)$ and
$D=(0, \infty)$. Finally, Q-monotonicity is also verified by
taking $Q(u+ \pi)=|u|$.

Now we shall prove that solutions $u$ to \eqref{oscillator eq},
above the separatrices in the plane phase and satisfying the
initial condition $u(0)=0$ are $(G'_R,G')$ equivariant for some
$R>0$ depending on $u'(0)$. For such solutions,
$\lambda:=u'(0)>2\sqrt{\alpha}$ holds, and
by integrating $u'u''=- \alpha (sinu)u'$, we obtain
\begin{equation}\label{oscillator speed}
u'=\sqrt{2W(u)+\lambda^2-4\alpha}\,.
\end{equation}
As a consequence, $u$ is strictly increasing with
$\lim_{x \to +\infty} u(x) = + \infty$. This ensures the existence
of a minimal $T>0$, depending on $\lambda$, such that $u(T)=2 \pi$.
Then, utilizing the periodicity of W, it is easy to see that
$u$ is $(G'_R,G')$ equivariant, for $R:=T/2\pi$.
Actually, since T is strictly decreasing from $+\infty$ to $0$,
when $\lambda \in (2\sqrt{\alpha},+ \infty)$, there exists for
every $R>0$ a unique $(G'_R,G')$ equivariant solution
to \eqref{oscillator eq}, called $u_R$.


According to Theorem \ref{thm1}, $u_R$ satisfies
$u_R((0,T/2))\subset (0,\pi)$, which is obvious, and
$|u_R(x)-\pi| \leq K \mathrm{e}^{-kx}$, for
$x \in F'_R=(0,T/2)$ and $R$ big enough. Considering then
the sequence $u_R:(-T/2,T/2)\rightarrow \mathbb{R}$, and passing
to the limit when $R \to \infty$, we capture the heteroclinic
solution $u$ of Theorem \ref{thm2}, which is  $G$ equivariant (i.e. odd)
and satisfies:
\begin{itemize}
\item[(i)] $u([0,\infty)) \subset [0,\pi]$;
\item[(ii)] $|u(x)-\pi| \leq K \mathrm{e}^{-kx}$,
for $x \in (0,+\infty)$.
\end{itemize}

In $\mathbb{R}^2$, we give other examples of potentials $W$
 satisfying our hypotheses for the discrete group $G'$ generated
by the reflections with respect to the lines $u_2=0$, $u_1=\pi$
and $u_1=u_2$. The fundamental alcove bounded by these three
lines is: $F'=\{(u_1,u_2)\in (0,\pi)\times(0,\pi) : u_1 > u_2 \}$.
Considering the auxilliary function $\phi (u)=\cos^2 u+\cos u+1/4$,
we construct the potentials:
$$
W_1(u_1,u_2)=\phi(u_1)+\phi(u_2),\quad
W_2(u_1,u_2)=\phi(u_1)+\phi(u_2)+\phi(u_1) \phi(u_2)
$$
which have a unique  minimum, nondegenerate, in $\overline{F'}$
at $a=(2\pi/3,2\pi/3)$. It can then be verified that $W_1$
and $W_2$ satisfy Q-monotonicity in $D'=(0,\pi)\times(0,\pi)$
with $Q(u)=|u-a|$, where $u=(u_1,u_2)$.

The Hypotheses (H1)--(H3) are exact analogs of the hypotheses
introduced in \cite{alikakos-fusco} for the point group $G$.
Actually our hypothesis on $Q$ here is less restrictive since
no symmetry assumptions are imposed (cf. \cite{alikakos}).
$Q$-monotonicity is a restrictive hypothesis on the potential.
We note that for $n=1$ and even symmetry, for a double-well
potential $W$, and $D=F=\{ u>0 \}$, $Q$-monotonicity implies
that for $u>0$, $W_{u}(u)(u-a_1) \geq 0$ holds. Nevertheless
$Q$-monotonicity allows for a large class of nontrivial
potentials (cf. \cite{alikakos-fusco}). Very recently,
in ongoing work,  Fusco (cf. \cite{fusco}) has been able to
remove the $Q$-monotonicity from the hypotheses in establishing
the main result in \cite{alikakos-fusco}. Since the main
contribution of the present work is the positivity result,
which does not require the $Q$-monotonicity hypothesis,
we decided not to take into account these new developments.

\section{Minimization}

Thanks to the proposition below, we can identify the class
of $(G'_R,G')$ equivariant Sobolev maps
$W^{1,2}_{\mathrm{E\; loc}}(R)(\mathbb{R}^n;\mathbb{R}^n)$,
with the class
\begin{equation}\label{K_R}
K^R:=\{u \in W^{1,2}(F'_R;\mathbb{R}^n) :
\mathrm{Tr}\: u(S_i^R)\subset \Pi_i\ , \forall i\,\,
  1 \leq i \leq N \}.
\end{equation}
of $W^{1,2}(F'_R;\mathbb{R}^n)$ maps such that the restriction
of the trace (denoted by $Tr$) to each of the faces
$S_i^R$ of $F'_R$, takes values almost everywhere in the
corresponding hyperplane $\Pi_i$.

\begin{proposition}\label{prop1}
The map which associates to each
$u \in W^{1,2}_{\mathrm{E\ loc}}(R)(\mathbb{R}^n;\mathbb{R}^n)$
its restriction to $F'_R$ is one-to-one and onto $K^R$.
\end{proposition}


\begin{proof}
We denote by $(x_1,\ldots,x_n)$ the coordinates of
$x \in \mathbb{R}^n$, and by $(u^1,\ldots,u^n)$ the components
 of a map u with values in $\mathbb{R}^n$. Without loss of
generality we assume that $R=1$ and that
$\Pi=\{(x_1,\ldots,x_n) \in \mathbb{R}^n : x_n=0 \}$
is the supporting plane of the face $S$ of $F'$.
Furthermore, we denote by $g(x_1,\ldots,x_n)=(x_1,\ldots,-x_n)$,
the reflection with respect to $\Pi$. To a map
$\phi:F'\to \mathbb{R}^n$, we associate its extension by reflection
to $gF'$, denoted by $\overline \phi$.
Clearly, we have for every $x \in gF'$:
\begin{equation}\label{bar}
\overline \phi(x):=g \phi (gx)=(\phi^1(x_1,\dots,-x_n),\dots,
\phi^{n-1}(x_1,\dots,-x_n),-\phi^n(x_1,\dots,-x_n)).
\end{equation}
It is easy to check that
$u \in W^{1,2}(F';\mathbb{R}^n) \Rightarrow \overline u
\in W^{1,2}(gF';\mathbb{R}^n)$. In addition, if the sequence
$\phi_m \in C^{\infty}(\overline{F'};\mathbb{R}^n)$ converges to
$u$ in $W^{1,2}(F';\mathbb{R}^n)$, then the sequence
$\overline {\phi_m} \in C^{\infty}(\overline{gF'};\mathbb{R}^n)$
also converges to $\overline u$ in $W^{1,2}(gF';\mathbb{R}^n)$.

Now let us consider a map
$u \in W^{1,2}_{\rm E\, loc}(\mathbb{R}^n;\mathbb{R}^n)$
and let us prove that $\operatorname{Tr} u(S) \subset \Pi$.
Obviously $u= \overline u$ and
$ (\operatorname{Tr} u) \big |_S = (\operatorname{Tr} \overline u)
\big |_S $, where $ \big | _S$ denotes the restriction of a map to $S$.
Then writing that
$ \lim_{m \to \infty} \phi_m \big |_S=\lim_{m \to \infty}
\overline{ \phi_m} \big |_S$ in $L^2(S;\mathbb{R}^n)$ and
utilizing \eqref{bar}, we find that
$ \lim_{m \to \infty} (\phi^n)_m \big |_S=0$
in $L^2(S,\mathbb{R})$ which means that $\operatorname{Tr} u(S) \subset \Pi$.
Therefore, the map of Proposition \ref{prop1}. takes its values
in $K^R$ and clearly by equivariance it is one-to-one.
To complete the proof, it remains to show that it is onto.

Indeed, if $u \in K^R$, we can extend it by reflection to a map
called $U$, defined in $\Omega:=
 \mathrm{Int} \left( \overline{F'} \cup g \overline{F'} \right)$,
setting:
\[
U(x) = \begin{cases}
u(x), &\text{for } u \in F',\\
\overline u(x) , &\text{for } u \in gF' .
\end{cases}
\]
Since $\operatorname{Tr} u(S) \subset \Pi$, it can be shown as
before that $ (\operatorname{Tr} \overline u^{ \: n}) \big |_S=0$ a.e..
Next, noting that by definition of $\overline u$,
$(\operatorname{Tr} u^i) \big |_S=(\operatorname{Tr}
\overline u^{ \: i}) \big |_S$ for $1 \leq i < n$,
we conclude that $U$ defines a $W^{1,2}(\Omega;\mathbb{R}^n)$
 map symmetric with respect to $\Pi$. Repeating this process,
$U$ can be extended to an equivariant map in the whole space.
\end{proof}

The Proposition above allows us to define for
$(G'_R,G')$ equivariant Sobolev maps the functional associated
to \eqref{system} by
\begin{equation}\label{action}
J_{F'_R}(u) = \int_{F'_R} \big\{ \frac{1}{2} |\nabla u|^2 + W(u) \big\}
 \mathrm{d} x,
\end{equation}
and to consider the minimization problem
\begin{equation}\label{actionb}
\min_{A^R} J_{F'_R}, \quad \text{where }
A^R := \big\{ u \in K^R : u(\overline{F'_R}) \subset \overline{F'}
\big\}.
\end{equation}
In the class $A^R$, we have imposed the \textit{positivity}
constraint:
\begin{equation}\label{positive}
u ( \overline{F'_R} ) \subset \overline{F'}.
\end{equation}
Note that the convexity of $F'$ implies that $A^R$ is convex
 and closed in $W^{1,2}(F'_R;\mathbb{R}^n)$, and so a
minimizer $u_R \in A^R$ exists.

\section{The gradient flow and positivity}

To show that the positivity constraint built in $A^R$ does not
affect the Euler-Lagrange equation we will utilize the gradient flow
\begin{equation}\label{ev}
\begin{gathered}
\frac{\partial u}{\partial t} = \Delta u - W_u(u) \quad
\text{in } \mathbb{R}^n \times (0,\infty),\\
u(x,0) = u_0(x) \quad \text{in } \mathbb{R}^n,
\end{gathered}
\end{equation}
with initial condition $u_0 \in W^{1,2}_{\rm E\, loc}
(R)(\mathbb{R}^n;\mathbb{R}^n) \cap L^{\infty}_{\rm loc}
(\mathbb{R}^n;\mathbb{R}^n)$.
Since $W$ is $C^3$ (cf.\ (H1)), the results in
\cite[Ch. 3, \S3.3, \S 3.5]{henry} apply and provide a
unique solution to \eqref{ev} in
$C([0,\infty);W^{1,2}_{\rm E\, loc}(R)(\mathbb{R}^n;\mathbb{R}^n))$.
For every bounded domain $\Omega \subset \mathbb{R}^n$,
this solution is for $t>0$, as a function of $x$, in
$C^{2+\alpha} (\overline{\Omega};\mathbb{R}^n)$, for some
$0<\alpha<1$. Moreover, if the initial condition $u_0$ is also
assumed globally Lipschitz in $\mathbb{R}^n$, the flow
$u \in C([0,\infty);C(\overline \Omega; \mathbb{R}^n))
\cap C(0,\infty;C^2(\overline \Omega; \mathbb{R}^n))$
for every bounded domain $\Omega$.


\begin{theorem}\label{thm3}
Let $W$ be a potential satisfying {\rm (H1)}.
If the initial condition $u_0 \in W^{1,2}_{\rm E\, loc}(R)
(\mathbb{R}^n;\mathbb{R}^n) \cap L^{\infty}_{\rm loc}
(\mathbb{R}^n;\mathbb{R}^n)$ is assumed to be positive
(cf. \eqref{positive}), then
\[
u(\cdot, t; u_0)(\overline{F'_R}) \subset \overline{F'}, \text{ for }
t \geq 0 \quad\text{(positivity),}
 \]
and, moreover,
\[
u(\cdot, t; u_0)(F'_R) \subset F' , \quad \text{for }t > 0\quad
\text{(strictly positivity)}
\]
\end{theorem}

\begin{proof}
Without loss of generality we assume $R=1$. We first prove the
Theorem when $u_0 \in
W^{1,2}_{\rm E\, loc}(R)(\mathbb{R}^n;\mathbb{R}^n)$
is globally Lipschitz, in which case $u$ is smooth and satisfies
for $t>0$ the Boundary Conditions \eqref{boundaryconditions}.
We shall give a detailed proof for $n=2$ and an outline in higher
dimensions $n\geq 3$, just pointing out the new elements.
Next, we shall consider the general case of an initial
condition
$u_0 \in W^{1,2}_{\rm E\, loc}(R)(\mathbb{R}^n;\mathbb{R}^n)
\cap L^{\infty}_{loc}(\mathbb{R}^n;\mathbb{R}^n)$.

\subsection*{Proof for $n=2$, and
$u_0 \in W^{1,2}_{\rm E\, loc}(R)(\mathbb{R}^2;\mathbb{R}^2)$
and globally Lipschitz}
In $\mathbb{R}^2$, $F'$ is either a triangle with acute angles
or a rectangle. We shall consider the case of a triangle,
the proof being similar for a rectangle. Let us denote by $P_0=O$,
 $P_1$ and $P_2$ the vertices of $F'$ and by  $S_1=P_0P_1$,
$S_2=P_0P_2$, $S_3=P_1 P_2$, its sides supposed to be closed.
$\Pi_1$, $\Pi_2$ and $\Pi_3$ are the corresponding supporting
lines of $S_1$, $S_2$, $S_3$, while for $1\leq i \leq 3$,
$r_i$ is the outer unit normal vector to the side $S_i$.
Finally, we also need to define the closed segment
$P_{ij}(t)P_{ji}(t):=u(P_iP_j,t)$ which is the image
of the side $P_iP_j$ by $u(.,t)$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2}
\end{center}
\caption{The fundamental alcove $F'$ and the extensions $P_{ij}P_{ji}$.}
\label{fig2}
\end{figure}

We shall suppose that there exists a $t_0>0$ for which
$u(\overline{F'},t_0) \not\subseteq \overline{F'}$, and seek
 a contradiction in the two following cases:

\noindent \textbf{Case 1:} $u(\partial F',t) \subset \overline{F'}$
for every $t\geq 0$.
Without loss of generality we can suppose that there
exists $x_0 \in F'$ such that for instance:
\begin{equation}\label{bar2}
\langle u(x_0,t_0)-P_0 \ , r_1\rangle >0 \,.
\end{equation}
Following \cite{alikakos-fusco}, we set
$h_1(x,t)=\langle u(x,t)-P_0 \ , r_1\rangle$ which is solution of
\begin{equation}\label{eqh1}
\frac{\partial h_1}{\partial t} = \Delta h_1
-\langle W_u(u) \ , r_1\rangle,
\end{equation}
and noting that
\begin{equation}\label{bar3}
\langle W_u(u),r_1 \rangle =  \langle
\int_0^1W_{uu}\big(u + (s-1)h_1 r_1 \big) r_1 \mathrm{d} s,\, r_1 \rangle h_1,
\end{equation}
due to the periodicity of $W$, \eqref{eqh1} can be written as
\begin{equation}\label{eqh1c}
\frac{\partial h_1}{\partial t}
= \Delta h_1 +c_1 h_1,
\end{equation}
 with $c_1$ continuous and bounded on
$\mathbb{R}^2 \times [0,\infty)$.
To have an equation with a non-positive first order coefficient,
we apply a well-known trick, considering instead of $h_1$
the function $\overline{h_1}(x,t)=e^{-\lambda t}h_1(x,t)$
which is solution to
$\frac{\partial \overline{h_1}}{\partial t}
= \Delta \overline{h_1} +(c_1-\lambda) \overline{h_1}$.
Choosing $\lambda \geq c_1$, the maximum principle applies to
$\overline{h_1}$ and gives
\begin{equation}\label{eqh1c2}
0<\max \big \{\overline{h_1}(x,t) : x \in \overline{F'},\,
 t \in[0,t_0] \big \}=\overline{h_1}(x_1,t_1)
\end{equation}
for some $x_1 \in F'$ and some $t_1 \in [0,t_0]$,
since we supposed that $h_1(x_0,t_0)>0$,
$h_1(\partial F',t) \subset (-\infty,0]$ for all $t \geq 0$
(hypothesis in case 1),
 and $h_1(\overline{F'},0) \subset (-\infty,0]$
(positivity of $u_0$).
As a consequence $\overline{h_1} \equiv \overline{h_1}(x_1,t_1)>0$
which contradicts the fact that $h_1(P_0,t) =0$,
for every $t \geq 0$.


\noindent\textbf{Case 2:}
$u(\partial F',t_2) \not\subseteq \overline{F'}$, for some $t_2> 0$.
This case is more difficult since we have to deal with the
extensions $P_{ij}(t)P_{ji}(t)$. As before, we consider
for $i=1,2,3$, the projection $h_i$ of $u$ with respect to the
vector $r_i$ which is solution to
$\frac{\partial h_i}{\partial t} = \Delta h_i +c_i h_i$,
with $c_i$ bounded. We will also need the functions
$\overline{h_i}(x,t)=e^{-\lambda t}h_i(x,t)$  which are solutions
to $\frac{\partial \overline{h_i}}{\partial t}
= \Delta \overline{h_i} +(c_i-\lambda) \overline{h_i}$,
and choose $\lambda \geq c_i$, for $i=1,2,3$.

Without loss of generality we can suppose that for instance
\begin{equation}\label{extension}
\max \big \{e^{-\lambda t}d(P_i,P_{ij}(t)) : t \in [0,t_2], 0\leq i,j
\leq 2, i \neq j \big \}=e^{-\lambda t_3}d(P_0,P_{01}(t_3)) \,,
\end{equation}
for some $t_3 \in (0,t_2]$.
By projecting $u$ onto the direction
$\tau:=- \frac{1}{| \overrightarrow{P_0P_1} |}
 \overrightarrow{P_0P_1}$ we define as before the functions
$h(x,t)=\langle u(x,t)-P_0 \ , \tau \rangle$ and
$\overline{h}(x,t)=e^{-\lambda t}h(x,t)$.
Since the angles of $F'$ are acute, $\tau$ can be written
$\tau=\alpha r_1+\beta r_2$ with $\alpha$, $\beta \geq 0$,
and therefore $h=\alpha h_1+\beta h_2$ and
$\overline h=\alpha \overline{h_1}+\beta \overline{h_2}$.
Clearly, $\frac{\partial{\overline h}}{\partial t}
= \Delta \overline h +\overline c \: \overline h$ holds
with $\overline c \leq 0$. In order to successfully apply
the maximum principle to $\overline h$, we note that
 by \eqref{extension} and since $F'$ has acute angles,
there exists a closed half-plane $E_{01}$ with the following
properties:
\begin{itemize}
\item[(i)] $e^{-\lambda t_3}P_{01}(t_3) \in \partial E_{01}$,
\item[(ii)] $\overrightarrow{P_0P_1} \; \bot \; \partial E_{01}$,
\item[(iii)] $e^{-\lambda t}P_{ij}(t) \in E_{01}$, for all
$t \in [0,t_2]$ and all $(i,j)$ such that $0\leq i \neq j \leq 2$.
\end{itemize}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{fig3}
\end{center}
\caption{The separating half-plane $E_{01}$ and the extensions
$P_{ij}P_{ji}$ according to the time}
\label{fig3}
\end{figure}

Now, let $\mu:=\max \big \{\overline{h}(x,t) :
x \in \overline{F'},\, t \in[0,t_2] \big \}
\geq e^{-\lambda t_3}d(P_0,P_{01}(t_3))>0$.
If $\mu>e^{-\lambda t_3}d(P_0,P_{01}(t_3))$, then necessarily
$\mu$ is attained at an interior point $x_2 \in F'$, for some
$t_4>0$. Thus for $x \in \overline{F'}$ and $t \in [0,t_4]$,
we must have $\overline{h}(x,t)=\mu$ which contradicts the fact
that $h(P_0,t) =0$, for every $t \geq 0$.
If $\mu=e^{-\lambda t_3}d(P_0,P_{01}(t_3))$, then there necessarily
exists an interior point $x_3$ of the segment $P_0P_1$ such that
$\overline h(x_3,t_3)=\mu$. In this case, we consider the extension
of $u$, $h$ and $\overline h$ to the union of two fundamental
alcoves $\Omega:= \mathrm{Int}
\left( \overline{F'} \cup g \overline{F'} \right)$
where $g$ denotes the reflection with respect to the line
$P_0P_1$. By equivariance of $u$,
$\mu=\max \big \{\overline{h}(x,t) : x \in \overline{\Omega}, \,
 t \in[0,t_2] \big \}$
also holds and since $x_3$ is an interior point of $\Omega$,
we reach the same contradiction as previously.

So far we have proved the first statement of the Theorem (positivity)
for smooth initial condition in $\mathbb{R}^2$.
To complete the proof in this case it remains to show that
$u(\cdot, t; u_0)(F') \subset F'$, for $t > 0$ (strong positivity).
To see this, suppose by contradiction that there exist $x_4 \in F'$
and $t_5>0$ such that $u(x_4,t_5) \in \Pi_1$ for instance; i.e.,
 $h_1(x_4,t_5)=0$. Then, $max \big \{h_1(x,t) : x \in \overline{F'},
\, t \in[0,t_5] \big \}=h_1(x_4,t_5)$ due to the positivity of $u$.
According to the maximum principle, this implies that for
$x \in \overline{F'}$ and $t \in [0,t_5]$, we have $h_1 (x,t)=0$
which contradicts the fact that $u(P_2,t) =P_2$, for every $t \geq 0$.

\subsection*{Proof for $n \geq 3$, and
$u_0 \in W^{1,2}_{\rm E\, loc}(R)(\mathbb{R}^n;\mathbb{R}^n)$
 and globally Lipschitz}
To prove the Theorem in higher dimension we proceed as
in $\mathbb{R}^2$. Essential is the fact that the angles between
the faces of $F'$ are acute (see \cite{humphreys}). This implies
in particular that any lower dimensional face has also acute angles.
 As before, we shall suppose that there exists a $t_0>0$ for which
$u(\overline{F'},t_0) \not\subseteq \overline{F'}$, and seek a
contradiction in the two following cases:

\noindent\textbf{Case 1:} $u(\partial F',t) \subset \overline{F'}$,
for every $t\geq 0$.
This case presents no difficulty. As previously,  to reach a
contradiction, we consider the projection:
$$
h_1(x,t)=\langle u(x,t)-P_0 \ , r_1\rangle,
$$
where $r_1$ is the
outer unit normal vector to the face $S_1$ and $P_0$ a
vertex of $S_1$.

\noindent\textbf{Case 2:}
$u(\partial F',t'_0) \not\subseteq \overline{F'}$, for some $t'_0> 0$.
Let $S_1$, $S_2$,\dots, $S_N$ be the closed faces of $F'$.
In the same way we considered in $\mathbb{R}^2$ the extensions
$P_{ij}(t)P_{ji}(t)$ of the sides of $F'$, in higher dimension
we have to deal with the extensions of its faces. Without loss
of generality and omitting the $e^{-\lambda t}$ term for reason
of simplicity, we can assume that for instance:
\begin{equation}\label{extension n dim}
\max \big \{d(u(x,t),S_{i}) : x \in S_i \: ,t \in [0,t'_0],
1\leq i \leq N \big \}=d(u(x_1,t_1),S_1)=\epsilon>0 \,,
\end{equation}
for some $x_1 \in S_1$ and $t_1 \in (0,t'_0]$.
Setting $u_1:=u(x_1,t_1)$, there exists since $S_1$ is compact
and convex, a unique point $v_1 \in \partial S_1$ such that
$d(u_1,v_1)=\epsilon$. We shall consider the projection $h$
of $u$ with respect to the vector
$\rho:= \frac{1}{| u_1 - v_1 |} (u_1 -v_1)$:
\begin{equation}\label{h,rho}
h(x,t)=\langle u(x,t)-v_1 , \rho \rangle .
\end{equation}
Since the angles of $F'$ are acute, the half-space
$E_0:=\big \{ x \in \mathbb{R}^n : \langle x-v_1 , \rho \rangle
\leq 0 \big \}$ contains $\overline {F'}$ and setting
$E_1:=\big \{ x \in \mathbb{R}^n : \langle x-u_1 , \rho \rangle
\leq 0 \big \}$ we have $d(\overline{F'},\partial E_1) \geq \epsilon$.
In particular, by \eqref{extension n dim}, $u(S_i,t) \subset E_1$,
for every $i=1,\dots,N$ and every $t \in [0,t'_0]$; that is,
the half-space $E_1$ contains all the extensions of the faces of $F'$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig4}
\end{center}
\caption{The separating half-space $E_1$}
\label{fig4}
\end{figure}

Now, let $\mu:=\max \big \{ h(x,t) : x \in \overline{F'}, \,
 t \in[0,t'_0] \big \} \geq h(x_1,t_1)=\epsilon$
and let us apply the maximum principle to $h$.
If $\mu>\epsilon$, then necessarily $\mu$ is attained at an
interior point of $F'$, and it is easy in this case to reach
a contradiction. Otherwise $\mu=\epsilon$ and the maximum of
$h$ is attained at $(x_1,t_1)$. We have the following hierarchy:
\begin{enumerate}
\item[(i)] if $\rho$ is not parallel to an $n-2$ dimensional face
 of $S_1$, then $x_1 \in Int(S_1)$;
\item[(ii)] if $\rho$ is parallel to an $n-2$ dimensional face of
 $S_1$ but not parallel to a lower dimensional face, then either
 $x_1 \in Int(S_1)$ or $x_1$ belongs to the interior of this $n-2$
 dimensional face \ldots
\item[(iii)] if $\rho$ is parallel to an edge of $S_1$, then $x_1$
belongs to the interior of this edge or to the interior of any
higher dimensional face of $S_1$ containing this edge.
\end{enumerate}

In all cases, if we consider the extension of $u$ and $h$ to
$\Omega:= \mathrm{Int}\big( {\cup_{g\in \Gamma} g\overline{F'}} \big)$
where $\Gamma:=G'_{x_1}$ is the stabilizer of $x_1$, we will have
 by equivariance
$$
\mu=\max \big\{h(x,t) : x \in \overline{\Omega},\, t \in[0,t'_0]
\big\}=h(x_1,t_1),
$$
and since $x_1$ is an interior point of $\Omega$ it is easy to
reach a contradiction. This proves the positivity of the gradient
flow for dimensions $n \geq 3$. The proof of strong positivity is
straightforward by applying the same arguments as on the plane.

\subsection*{Proof for a general $u_0 \in W^{1,2}_{\rm E\, loc}(R)
(\mathbb{R}^n;\mathbb{R}^n) \cap L^{\infty}_{\rm loc}
(\mathbb{R}^n;\mathbb{R}^n)$}
The idea is to approximate $u_0$ by smooth positive equivariant
maps an then utilize the continuous dependence for the flow of
initial condition. We need the following
result\footnote{In a former version of \cite{alikakos-fusco}
such an argument appeared.}.

\begin{proposition}\label{prop2}
If $u \in W^{1,2}_{\rm E\, loc}(R)(\mathbb{R}^n;\mathbb{R}^n)$
is such that $u ( \overline{F'_R} ) \subset \overline{F'}$
then there exists a sequence $(u_m) \subset
C^{\infty}(\overline{F'_R};\mathbb{R}^n)$ with the following
 properties:
\begin{itemize}
\item[(i)] $(\operatorname{Tr} u_m)(S_i^R) \subset \Pi_i$,
  for every face $S_i^R$ of $F'_R$,
\item[(ii)] $u_m( \overline{F'_R} ) \subset \overline{F'}$,
\item[(iii)] $u_m$ converges to $u$ in $W^{1,2}(F'_R;\mathbb{R}^n)$
  as $m \to \infty$.
\end{itemize}
\end{proposition}


\begin{proof}
Without loss of generality we can suppose that $R=1$. In what follows
we will utilize the standard mollifier 
$\eta_m(x):=m^n (\int \eta )^{-1} \eta(mx)$, $m \geq1$, where
\[ 
\eta(x) =  \begin{cases}
e^{(|x|^2-1)^{-1}} &\text{for } x \in \mathbb{R}^n, \;  |x|<1,\\
0  &\text{for } x \in \mathbb{R}^n, \;  |x| \geq 1 .
\end{cases} 
\]
Let us consider the restriction of u to a smooth domain $\Omega$ containing
$\overline{F'}$ and let us extend it to a  
$W^{1,2}(\mathbb{R}^n;\mathbb{R}^n)$ map called $\overline u$. 
Setting $\overline{u_m}:=\overline u \ast \eta_m$, it is clear that  
$\overline{u_m} \in C^{\infty}(\mathbb{R}^n;\mathbb{R}^n) 
\cap W^{1,2}(\mathbb{R}^n;\mathbb{R}^n)$ and that $\overline {u_m}$ 
converges to $\overline u$ in $W^{1,2}(\mathbb{R}^n;\mathbb{R}^n)$ 
as $m \to \infty$. Therefore, if we define $(u_m)$ as the restrictions of 
$(\overline{u_m})$ to $F'$, (iii) will be trivially satisfied. 
Before proving (i) and (ii), we need to introduce some notation. 

Let  us denote by $S$ a face of $F'$, by $r$ the outer unit normal vector 
to $S$, by $\Pi=\{x \in \mathbb{R}^n  :  \langle x ,r \rangle=k\}$ 
the supporting plane of $S$, and  by $g$ the reflection with respect to $\Pi$. 
$\Pi$ defines the closed half-spaces $E^-$ and 
$E^+=\{x \in \mathbb{R}^n  : \langle x ,r \rangle \leq k\}$ 
which contains $F'$.

For $x \in S$, we compute
\begin{align*}
u_m(x) 
&= \int_{B(x,1/m)} \eta_m(x-y)u(y) \,\mathrm{d}y \\
&=\int_{B(x,1/m)  \cap E^+} \eta_m(x-y)u(y) \,\mathrm{d}y
 +\int_{B(x,1/m) \cap E^-} \eta_m(x-z)u(z) \,\mathrm{d}z  \\
&= \int_{B(x,1/m) \cap E^+} \eta_m(x-y)u(y) \,\mathrm{d}y
 +\int_{B(x,1/m) \cap E^+} \eta_m(gx-gy)u(gy) \,\mathrm{d}y\\
&= \int_{B(x,1/m) \cap E^+} \eta_m(x-y)(u(y)+gu(y)) \,\mathrm{d}y 
\end{align*}
and from this expression we see that 
\[
\langle u_m(x) ,r \rangle  
= \int_{B(x,1/m) \cap E^+} \eta_m(x-y) 2k \mathrm{d}y =k  \quad 
\text{(by equivariance of u)} 
\]
which means that $(u_m)$ satisfies (i). To prove (ii), take 
$x \in \overline{F'}$ and compute as before 
\begin{align*}
u_m(x) 
&= \int_{B(x,1/m)} \eta_m(x-y)u(y) \,\mathrm{d}y  \\
&= \int_{B(x,1/m) \cap E^+} \eta_m(x-y)u(y) \,\mathrm{d}y
 +\int_{B(x,1/m) \cap E^-} \eta_m(x-z)u(z) \,\mathrm{d}z  \\
&= \int_{B(x,1/m) \cap E^+} \eta_m(x-y)u(y) \,\mathrm{d}y
 +\int_{B':=g(B(x,1/m) \cap E^-)} \eta_m(x-gy)gu(y) \,\mathrm{d}y \\
&= \int_{(B(x,1/m) \cap E^+) \setminus B'} \eta_m(x-y)u(y) \,\mathrm{d}y 
 +\int_{B'} (\eta_m(x-y)-\eta_m(x-gy))u(y) \,\mathrm{d}y \\
& \quad +\int_{B'} \eta_m(x-gy)(u(y)+gu(y)) \,\mathrm{d}y  \,.
\end{align*}
From this decomposition, the equivariance of $u$, the fact that 
$u(E^+) \subset E^+$ and the properties of the mollifier, we deduce that 
\begin{align*}
\langle u_m(x) ,r \rangle  
& \leq k \int_{(B(x,1/m) \cap E^+) \setminus B'}\eta_m(x-y) \,\mathrm{d}y 
 + k \int_{B'}(\eta_m(x-y)-\eta_m(x-gy)) \,\mathrm{d}y  \\
& \quad +2k \int_{B'} \eta_m(x-gy) \,\mathrm{d}y 
\end{align*}
and
\begin{align*}
\langle u_m(x), r \rangle 
& \leq k  \int_{(B(x,1/m) \cap E^+) \setminus B'} \eta_m(x-y) \,\mathrm{d}y 
 + k \int_{B'}  \eta_m(x-y) \,\mathrm{d}y  \\
&  -k \int_{B(x,1/m) \cap E^-} \eta_m(x-y) \,\mathrm{d}y 
 +2k \int_{B(x,1/m) \cap E^-} \eta_m(x-y) \,\mathrm{d}y 
\end{align*}
Thus $\langle u_m(x) ,r \rangle \leq k$ which means that $u(x) \in E^+$ 
and completes the proof. 
\end{proof}

Taking into account Propositions \ref{prop1} and \ref{prop2},
it is possible to construct for every initial condition
$u_0 \in W^{1,2}_{\rm E\, loc}(R)(\mathbb{R}^n;\mathbb{R}^n)$
 a sequence of positive maps
$(u_m)\subset W^{1,2}_{\rm E\, loc}(R)(\mathbb{R}^n;\mathbb{R}^n)$,
globally Lipschitz in $\mathbb{R}^n$ (and actually $C^{\infty}$
in the closure of every fundamental alcove) which converges to $u_0$
in $W^{1,2}_{loc}(\mathbb{R}^n;\mathbb{R}^n)$.
Utilizing then the continuous dependence for the flow of the initial
condition one can therefore prove the positivity results
of Theorem \ref{thm3}. in the general case.
\end{proof}

\section{Proofs of Theorem \ref{thm1} and Theorem \ref{thm2}}

In this section we follow \cite{alikakos}.
By taking for initial condition in Theorem \ref{thm3}
the minimizer $u_R$ constructed in Section 4, we have
$u(\cdot,t;u_R) \in A^R$, for $t\geq 0$. Since $u_R$ is a
global minimizer of $J_{F'_R}$ in $A^R$, and since
$u(\cdot,t;u_R) \in C^1(0,\infty;C^{2+\alpha} (\overline{F'_R}))$,
is a classical solution to \eqref{ev} for $t>0$, we conclude from
\begin{equation}\label{j-time}
\frac{\mathrm{d}}{\mathrm{d} t} J_{F'_R} (u(\cdot,t))
= - \int_{F'_R} |u_t|^2 \,\mathrm{d} x
\end{equation}
that $|u_t (x,t)|=0$, for all $x \in \mathbb{R}^n$ and $t>0$.
Hence, for $t>0$, $u(\cdot, t)$ is satisfying
\begin{equation}\label{u-satisfies}
\Delta u(x,t) - W_u (u(x,t)) = 0.
\end{equation}
By taking $t \to 0+$ and utilizing the continuity of the flow in
$W^{1,2} (F'_R;\mathbb{R}^n)$ at $t=0$,
$u(\cdot,\cdot;u_R) \in C([0,\infty);W^{1,2}(F'_R; \mathbb{R}^n))$,
we obtain that $u_R$ is a $(G'_{R},G')$ equivariant classical
solution to system \eqref{system} satisfying also
$u_R( F'_R)\subset F'$.

To prove the estimate in the second statement of
Theorem \ref{thm1} we need the following lemma.

\begin{lemma}\label{lemmatechnic}
Let $u_R$ be as above,
$H':=\mathrm{Int}\big( {\cup_{g\in G}} \overline{F'} \big)$
and $H'_R:=\mathrm{Int}\big( {\cup_{g\in G}} \overline{F'_R} \big)$.
Then for $R>1$ the following hold:
\begin{itemize}
\item[(i)] $\| u_R \|_{L^{\infty}(H'_R; \mathbb{R}^n)} \leq M$,
 where $M:= \max_{u \in \overline{H'}} |u|$,
\item[(ii)] $Q(u_R(x)) \leq \overline{Q}$, for $x \in H'_R$, where
 $\overline{Q} := \max_{u \in \overline{H'}} Q(u)$,
\item[(iii)] $J_{F'_R} (u_R) \leq C R^{n-1}$, where $C$ is a positive
 constant independent of $R$,
\item[(iv)] $\Delta Q(u_R (x)) \geq 0$, in
$W_{\rm loc}^{1,2} (D'_R;\mathbb{R}^n)$, where $D'_R$ is as in
Theorem \ref{thm1}.
\end{itemize}
\end{lemma}

\begin{proof}
(i) and to (ii) are trivial. For (iii), define
\[
b_{R}(x) :=
\begin{cases}
((x/R)-a)(1-d(x;\partial F'_R))+a, &\text{for }
 x \in \overline{F'_{R}} \text{ with } d(x;\partial F'_R) \leq 1,\\
a,  &\text{for } x \in F'_{R} \text{ with }
 d(x;\partial F'_R) \geq 1.
\end{cases}
\]
$b_R$ is continuous and $C^{\infty}$ piecewise  in
$\overline{F'_R}$ thus it defines a $W^{1,2}(F'_R;\mathbb{R}^n)$
map which also satisfies $b_R(\overline{F'_R}) \subset \overline{F'}$,
by convexity of $F'$. According to Proposition \ref{prop1}
it can be extended equivariantly on the whole space since for
$x \in S_i^R$, we have $b_R(x)=x/R \in S_i$. An easy computation
shows that $|\nabla b_R|$ and $W(b_R)$ are bounded in $F'_R$,
independently of $R$, and clearly these quantities vanish when
$d(x;\partial F'_R) \geq 1$. This implies that
$J_{F'_R} (b_R) \leq C R^{n-1}$ and since
$J_{F'_R} (u_R) \leq J_{F'_R} (b_R)$ by definition of $u_R$,
we obtain the desired estimate.

For (iv), see \cite[Lemma 4.1]{alikakos}.
\end{proof}

Estimate (iii) in Lemma \ref{lemmatechnic} and the subharmonicity
of $Q(u_{4R})$ in $D'_{4R}$ allow us to obtain via an iterated
application of the \textit{De Giorgi oscillation Lemma}
 the following pointwise estimate, for $R \geq R_0$:
\begin{equation}\label{Q estimate}
\sup_{B(x_R,R^*)} Q(u_{4R}(x)) \leq \bar{q},
\end{equation}
where $R^* = {R} / {2^k}$ (with $k$ integer independent of $R$),
$x_R:=4Rx_0$ (with $x_0 \in D'$ fixed), and $\overline q$
as in Hypothesis 2.
Utilizing then the comparison arguments in
\cite[Section 5]{alikakos},
it is possible to show that the ball
$B(x_R,R^*)$ in \eqref{Q estimate} can be replaced by a large
set $D'_{R^*}$ which includes all of $D'_{4R}$ with the exception of
a strip along the boundary $\partial D'_{4R}$ of width $d_0$
independent of $R$, for $R \geq R_0$; that is,
\begin{equation}\label{drstar-supset}
D'_{R^*} \supset \{ x \in D'_{4R} : d(x,\partial D'_{4R}) \geq d_0 \}.
\end{equation}
This result states that the minimizer $u_{4R} (x)$ on a set of
large measure is close to $a$, the zero of $W$ in $D'$,
for $R \to \infty$. From it, by applying, once again comparison
arguments, follows the estimate of Theorem \ref{thm1}:
\begin{equation}
|u_R(x)-a| \leq K \mathrm{e}^{-k d(x,\partial D'_R)}
\end{equation}
which holds for $R>R_0$ and $x \in D'_R$.
Finally, the uniform bound (i) provided by Lemma \ref{lemmatechnic}
and elliptic regularity, via a diagonal argument, allow us
to pass to the limit along a subsequence in $R$ and capture a
function
\begin{equation}\label{cv unif}
u(x) = \lim_{R_{n} \to \infty} u_{R_n}\big| _{H'_{R_n}},
\end{equation}
where $ \big| _{H'_{R}}$ denotes the restriction of a map to
$H'_R=\mathrm{Int}\big( {\cup_{g\in G}} \overline{F'_R} \big)$.
Since the convergence in \eqref{cv unif} is uniform up to the second
derivatives on compact sets, one can then see that the limit $u$
satisfies the exponential estimate of Theorem \ref{thm2} and
is also a $G$ equivariant solution to
\[
\Delta u -W_u(u) = 0\quad \text{in } \mathbb{R}^n.
\]
Clearly, positivity is verified
(i.e. $u(\overline{F})\subset \overline{F'}$)
and with the help of the exponential estimate, strong positivity
can be established in $D'$, that is $u(D)\subset D'$.

\subsection*{Acknowledgments}
We would like to thank Giorgio Fusco for introducing us to
the lattice set up by sharing his notes, and Rafe Mazzeo for
several discussions related to the regularity issue for domains
with corners.

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\end{document}