text:
\begin{document}

\title{On the Allen-Cahn
equation in the Grushin plane:\\
a monotone entire solution\\
that is not one-dimensional}

\author{\textit{Isabeau Birindelli}\ \ \& \ \
\textit{Enrico Valdinoci}}

\maketitle

\begin{abstract}
We consider
solutions of the
Allen-Cahn
equation in the whole Grushin plane
and we show that if they are monotone
in the vertical direction, then they are stable
and they satisfy a good energy estimate.

However, they are not necessarily
one-dimensional, as a counter-example shows.
\end{abstract}

\section{Introduction}

We consider here the
Grushin plane $G$ (see~\cite{Grushin}),
that is $\R^2$ endowed
with
the vector fields $X=\frac{\partial}{\partial x}$ and $Y
=x\frac{\partial}{\partial y}$.
We also define $T:=[X,Y]=\frac{\partial}{\partial y}$.

The Grushin gradient is then $\nabla_G:=(X,Y)$
(with the coordinates taken in the $(X,Y)$-frame)
and the Grushin Laplacian is $\Delta_G:=X^2+Y^2$.

We denote by $<\cdot,\cdot>$ the standard
scalar product
(when the vectors are taken in the $(X,Y)$-frame),
so that, for a smooth function $v$ we have
$$ |\nabla_G v(\zeta)| = \sqrt{<\nabla_G v(\zeta),
\nabla_G v(\zeta)>}=\sqrt{
(Xv(\zeta))^2+(Yv(\zeta))^2
},$$
for any $\zeta\in\R^2$.

Moreover, the Grushin norm
on $G$ is defined as
$$
\| (x,y)\|:=\sqrt[4]{|x|^4+4|y|^2}
$$
for any $(x,y)\in\R^2$, and then the Grushin ball
of radius $R> 0$ centered at $\zeta \in\R^2$ is
$$ B_R (\zeta):= \{ \eta\in\R^2
\st \|\eta-\zeta\|<R\}.$$

The main purpose of this paper is to
study
solutions of the
Allen-Cahn
equation in the Grushin plane, that is
\begin{equation}\label{main pde}
\Delta_G u(\zeta) + f(u(\zeta))=0\end{equation}
for any $\zeta\in\R^2$.

We take, for simplicity, $f\in C^1$,
though less regularity is also possible to be dealt with.

A particular case of interest
is when $f=-W'$ and $W$ is a double-well potential.
Namely, through this paper,
we denote by $W$ a function with the following properties:
$W\in C^{2} (\R)$ is
an even function for which $W(\pm 1)=0\leq W(r)$
for any $r\in\R$, $W''(0)\ne 0$, $W''(\pm 1)\ne0$,
and such that
\begin{equation}\label{UN}
{\mbox{
$W'(s)=0$ if and only if $s\in \{-1,0,+1\}$.
}}
\end{equation}
\bigskip

Inspired by a famous conjecture of De Giorgi (see
\cite{DG}), one may wonder under which conditions
the solutions of \eqref{main pde}
are one-dimensional, i.e.,
their level sets are straight lines and so,
up
to rotation, they depend on only one variable (at least when
$f=-W'$).

Natural requirements for such symmetry
are monotonicity and stability
conditions.
Namely,
if $u$ is a solution of \eqref{main pde},
we say that $u$ is stable if
\begin{equation}\label{Stab}
\int_{\R^2} |\nabla_G \phi(x)|^2-\int_{
{\mathcal{G}}_u}
f'(u(x)) \big(\phi(x)\big)^2
\,dx\ge0
\end{equation}
for any $\phi\in C^\infty_0(\R^2)$.

Stability is a natural condition in the calculus
of variation, since it states that the energy
functional associated to \eqref{main pde} has
non-negative second derivative. The stability condition
has thus been widely used in connection with the
problems posed by \cite{DG} (see, for instance, \cite{AAC, FSV}
and references therein).

Also, in the Euclidean setting, the stability condition
holds true whenever $u$ is monotone in some direction.
The analogy in the Grushin setting is somehow more
delicate, since the space is not homogeneous with respect
to the choice of a particular direction.

Thus, the monotonicity studied in this paper is the following.
We are mostly concerned with solutions that are monotone
in the $y$-direction, that is for which
\begin{equation}\label{Mo}
T u(\zeta)>0 \,{\mbox{for any}}\,
\zeta\in\R^2.
\end{equation}
We shall show that \eqref{Mo} implies~\eqref{Stab}
(see Proposition \ref{677888188} below).
\bigskip

Symmetry properties for
solutions of \eqref{main pde}
in the Grushin plane
have been recently studied
in \cite{VF}.
For instance, \cite{VF} pointed out the following result:

\begin{theorem}
Let us assume that $u$ is a stable solution of
\eqref{main pde}
in the whole $\R^2$ such that
$$ TYuXu(\zeta)-
TXuYu(\zeta)\leq 0\qquad{\mbox{for any $\zeta
\in\R^2$}}.
$$
Suppose that there exists $C_o\geq 1$ in such a way that
\begin{equation}\label{enesty}
\int_{B(0,R)}x^2| \nabla_G u|^2 \leq C_o R^4,
\end{equation}
for any $R\ge C_o$.

Assume also that
\begin{equation}\label{56789999900}
\nabla_G u (\zeta)\ne 0\,{\mbox{ for any }}\,\zeta\in\R^2.
\end{equation}
Then, $u$ depends only on the $x$-variable.
\end{theorem}

We observe that \eqref{56789999900}
is also a
sort of monotonicity condition, while
\eqref{enesty} is an energy growth requirement
(and energy bounds are often needed in the Euclidean
case too, see \cite{AAC}).
We shall show that \eqref{Mo} implies also
\eqref{enesty}, at least when $f=-W'$
(see Theorem \ref{EB} below). \bigskip

This said, a natural question arises.
Namely, \begin{question}\label{QQQ}
Is it true that bounded solutions
of \eqref{main pde} which satisfy \eqref{Mo}
are one-dimensional (at least for $f=-W'$)?
\end{question}

Note that one may be quite tempted to answer yes to such
a question, since \eqref{Mo} implies both
the stability condition and the good energy growth
in \eqref{enesty} (again, see for this
Proposition \ref{677888188}
and Theorem \ref{EB} here below).

The main purpose of this paper is in fact to show
that the above question has a \emph{negative
answer}.

This will be accomplished in Theorem \ref{MMM},
by constructing a counter-example which follows the
lines of the one in \cite{BL}.
\bigskip

The paper is organized in the following way.
After gathering some elementary observations in
Section \ref{S1}, we
point out in Section \ref{S0} that the monotonicity
condition in \eqref{Mo} implies the stability condition
in \eqref{Stab}.

Then, we develop in Section \ref{S2}
the energy estimates which show
that the monotonicity
condition in \eqref{Mo} also implies
the energy growth in \eqref{enesty}.

Finally, Section \ref{S3} contains the construction
of the counter-example which shows that Question \ref{QQQ}
has a negative answer.

\section{Preliminaries}\label{S1}

We collect in this section some elementary, but
useful, observations. The expert reader may surely
skip this section.

\subsection{An integration by parts}

We now point out a variation of Green formula,
complicated here by the non-ho\-mo\-ge\-ne\-ous
Grushin scaling.

\begin{lemma}\label{Green}
Let $u\in \Lambda^2(\R^2)$, $v\in
\Lambda^1(\R^2)$. Suppose that $|\nabla_G u|\in
L^\infty(\R^2)$ and that
$v\ge 0$.

Then
\begin{equation}\label{1ai889}
\begin{split}
& \int_{B_R(0)}\Big(<\nabla_G u,\nabla_G v>
+\Delta_G u v\Big)\\& \qquad\qquad \geq
-R^2\, \| \nabla_G u\|_{L^\infty(\R^2)}
\int_{\partial B_1(0)} v(RX, R^2Y)\,d\cH^1(X,Y).
\end{split}\end{equation}
\end{lemma}

\begin{proof} We set $U(X,Y):=u(RX,R^2Y)$,
$V(X,Y):=v(RX,R^2Y)$.
Then, by changing variable, we have
\begin{eqnarray*}
&& \int_{B_R(0)}\Big(<\nabla_G u,\nabla_G v>
+\Delta_G u v\Big)\\
&=&
\int_{B_R(0)}\Big( \partial_x u(x,y)\partial_x v(x,y)
+x^2
\partial_y u(x,y)\partial_y v(x,y)
\\ &&
+ \partial_{xx}u(x,y) v(x,y)
+x^2 \partial_{yy}u(x,y) v(x,y)
\Big)\,dx\,dy\\
&=&\frac{1}{R^2}
\int_{B_R(0)}\Big( \partial_X U
\Big(\frac{x}{R},\frac{y}{R^2}\Big)
\partial_X V
\Big(\frac{x}{R},\frac{y}{R^2}\Big)
\\ &&+
\frac{x^2}{R^2}
\partial_Y U
\Big(\frac{x}{R},\frac{y}{R^2}\Big)
\partial_Y V
\Big(\frac{x}{R},\frac{y}{R^2}\Big)
+ \partial_{XX}U\Big(\frac{x}{R},\frac{y}{R^2}\Big)
V\Big(\frac{x}{R},\frac{y}{R^2}\Big)
\\&&+\frac{x^2}{R^2} \partial_{YY}U
\Big(\frac{x}{R},\frac{y}{R^2}\Big)V
\Big(\frac{x}{R},\frac{y}{R^2}\Big)
\Big)\,dx\,dy
\\ &=& R\int_{B_1(0)}
\partial_X U(X,Y)\partial_X V(X,Y)+X^2
\partial_Y U(X,Y)\partial_Y V(X,Y)\\&&+
\partial_{XX} U(X,Y)V(X,Y)+
X^2
\partial_{YY} U(X,Y) V(X,Y)\,dX\,dY.
\end{eqnarray*}
Thence, by the standard Euclidean Divergence Theorem,
\begin{eqnarray}\label{8uauiiiq}
\nonumber
&& \int_{B_R(0)}\Big(<\nabla_G u,\nabla_G v>
+\Delta_G u v\Big)\\
&=& R \int_{B_1(0)}
{\rm div}\, \Big[V(X,Y) \Big(
\partial_X U(X,Y),
X^2 \partial_Y U(X,Y)\Big) \Big]
\,dX\,dY\\
&=& R
\int_{\partial B_1(0)}
V(X,Y) \Big(
\partial_X U(X,Y),
X^2 \partial_Y U(X,Y)\Big) \cdot
\nu^E (X,Y)
\,d\cH^1(X,Y),\nonumber
\end{eqnarray}
where
``$\cdot$'' denotes the standard Euclidean scalar product
and ``$\nu^E$'' is the standard Euclidean outward normal
of $\partial B_1(0)$.

We write \eqref{8uauiiiq} as
\begin{eqnarray*}
&& \int_{B_R(0)}\Big(<\nabla_G u,\nabla_G v>
+\Delta_G u v\Big)\\
&=& R^2
\int_{\partial B_1(0)}
v(RX,R^2 Y) \Big(
\partial_x u(R X,R^2 Y),
R X^2 \partial_y u(RX,R^2Y)\Big)\\&&
\qquad\qquad \cdot
\nu^E (X,Y)
\,d\cH^1(X,Y)
\end{eqnarray*}
and so, since the Euclidean norm of $\nu^E$ is $1$,
\begin{equation}\label{ai889}
\begin{split}
&\int_{B_R(0)}\Big(<\nabla_G u,\nabla_G v>
+\Delta_G u v\Big)\\
&\qquad \geq - R^2
\int_{\partial B_1(0)}
v(RX,R^2 Y) \Big| \Big(
\partial_x u(R X,R^2 Y),
R X^2 \partial_y u(RX,R^2Y)\Big)
\Big|_E
\,d\cH^1(X,Y),
\end{split}\end{equation}
where $|\cdot|_E$ is the Euclidean norm.

We now observe that, for any $(X,Y)\in\partial B_1(0)$,
we have $|X|\le 1$ and
\begin{eqnarray*}
&& \Big| \Big(
\partial_x u(R X,R^2 Y),
R X^2 \partial_y u(RX,R^2Y)\Big)\Big|_E^2
\\ &=&
\Big(
\partial_x u(R X,R^2 Y)\Big)^2+
R^2 X^4 \Big(\partial_y u(RX,R^2Y)\Big)^2
\\ &\le&
\Big(
\partial_x u(R X,R^2 Y)\Big)^2+
R^2 X^2 \Big(\partial_y u(RX,R^2Y)\Big)^2
\\ &=&
|\nabla_G u (RX,R^2 Y)|\\ &\leq&
\| \nabla_G u\|_{L^\infty(\R^2)}
.
\end{eqnarray*}
{F}rom this and \eqref{ai889} we get \eqref{1ai889}.
\end{proof}

\subsection{An interpolation inequality}

We point out the following elementary estimate:

\begin{lemma}
Let $h\in C^2 (\R)$.
Then,
\begin{equation}\label{Interpo}
\| h' \|_{L^\infty (\R)}\le
2\Big(\| h \|_{L^\infty (\R)}+
\| h'' \|_{L^\infty (\R)} \Big).\end{equation}
\end{lemma}

\begin{proof} We may assume that both~$\| h \|_{L^\infty (\R)}$
and~$\|
h'' \|_{L^\infty (\R)}$ are finite, otherwise~\eqref{Interpo}
is void.

First, we observe that, for any~$j\in\Z$,
there exists~$t_j \in [j,j+1]$ in such a way that
\begin{equation}\label{st1}
|h'(t_j)|\le 2\| h \|_{L^\infty (\R)}.
\end{equation}
Indeed, if~\eqref{st1} were false, there would exist~$j_o
\in\Z$ such that~$|h'(t)|> 2\| h \|_{L^\infty (\R)}$
for any~$t\in [j_o, j_o+1]$. Since~$h'$
is continuous, this means that either~$h'(t)>
2\| h \|_{L^\infty
(\R)}$ or~$
h'(t)<- 2\| h \|_{L^\infty (\R)}$
for any~$t\in [j_o,j_o+1]$. We assume that
the second possibility holds (the first case
is analogous). Then,
\begin{eqnarray*}&&
-2\| h \|_{L^\infty (\R)}
\leq h(j_o+1)-h(j_o)=
\int_{j_o}^{j_o+1} h'(t)\,dt \\&&
\qquad <
\int_{j_o}^{j_o+1}
\big(-
2\| h \|_{L^\infty (\R)}\big)
\,dt=-2\| h \|_{L^\infty (\R)}.
\end{eqnarray*}
This contradiction proves~\eqref{st1}.

Then, making use of~\eqref{st1},
given any~$j\in\Z$ and any~$t\in [j,j+1]$,
\begin{equation*}\begin{split}
&|h'(t)|\le |h'(t_j)|+\left| \int_{t_j}^t
h''(s)\,ds\right|\\
&\qquad\le
2\| h \|_{L^\infty (\R)}+
\| h '\|_{L^\infty (\R)} |t-t_j|\le
2\| h \|_{L^\infty (\R)}+
\| h' \|_{L^\infty (\R)}.
\qedhere
\end{split}\end{equation*}
\end{proof}

\subsection{ODE analysis}\label{odes}

The scope of this section is an elementary analysis
of the solutions $h\in C^2(\R)$ of
\begin{equation}\label{in2}
h''(t)=W'(h(t)) \qquad\qquad{\mbox{for
any $t\in\R$.}}\end{equation}

Recall that for any any $C^2$ solution of (\ref{in2}) and any
 $s$, $t\in\R$,
 \begin{equation} \label{intepri} \frac{|h'(s)|^2}{2}-W(h(s))=
\frac{|h'(t)|^2}{2}-W(h(t)).
\end{equation}

Furthermore
\begin{lemma}
Let $h$ be bounded. Then,
\begin{equation}\label{17889}
\| h\|_{C^2(\R)} \leq C,\end{equation}
for a suitable $C>0$, possibly depending on $\|h\|_{
L^\infty(\R)}$.

Also, for any $t\in\R$,
\begin{equation}\label{7891}
-W\Big(\inf_\R h\Big)=-W\Big(\sup_\R h\Big)=
\frac{|h'(t)|^2}{2}-W(h(t)).
\end{equation}
\end{lemma}

\begin{proof} By construction,
$$|h''(t)|\leq \max_{[-\| h\|_{L^\infty(\R)},
\| h\|_{L^\infty(\R)}]} |W'|$$
and so, by \eqref{Interpo},
we get \eqref{17889}.

We take
$$ \sigma \in\Big\{ \inf_\R h,
\sup_\R h\Big\}.$$
Let also $t_n$ be a sequence for which
$$ \lim_{n\rightarrow+\infty} h(t_n)=\sigma.$$
Let $w_n(t):= h(t+t_n)$. {F}rom \eqref{17889},
we have that $w_n$ converges, up to subsequence,
in $C^1_{\rm loc}(\R)$ to some function $w\in C^1(\R)$.

We now suppose that
$\sigma = \inf_\R h$ (for this argument, the
case $\sigma=\sup_\R h$
is completely analogous). Then,
$$ w(0)=
\lim_{n\rightarrow+\infty} w_n(0)=
\lim_{n\rightarrow+\infty} h(t_n)=
\sigma \le \lim_{n\rightarrow+\infty} h(t+t_n)
= w(t)$$
for any $t\in \R$, so $w'(0)=0$ and therefore,
by
(\ref{intepri}), for any $t\in\R$,
\begin{equation*} \begin{split}
& \frac{|h'(t)|^2}{2}-W(h(t))
=\lim_{n\rightarrow+\infty}
\frac{|h'(t_n)|^2}{2}-W(h(t_n))
\\ &\quad
=\lim_{n\rightarrow+\infty}
\frac{|w'_n(0)|^2}{2}-W(w(0))
= -W(\sigma).\qedhere\end{split}\end{equation*}
\end{proof}

\begin{lemma}\label{lepa}
If $h'(t_o)=0$, then $h$ is symmetric  with respect to $t=t_o$,
that is $h(t_o-t)=h(t_o+t)$ for any $t\in\R$.
\end{lemma}

\begin{proof} We set $h_\pm (t):=h(t_o\pm t)$.
Since $h_\pm ''(t)= W'(h_\pm(t))$ for any $t\in\R$,
$h_\pm (0)=h(t_o)$ and $h_\pm'(0)=0$, we deduce
from Cauchy Uniqueness Theorem that $h_+(t)=h_-(t)$.
\end{proof}

\begin{lemma}\label{2orm}
If $h$ has two or more critical points, then it
is periodic.
\end{lemma}

\begin{proof}
Suppose that $h'(a)=h'(b)=0$ with $b>a$ and let $T:=b-a$.
Then, utilizing Lemma \ref{lepa},
\begin{equation*}\begin{split}&
h(t+T)=h(b+(t-a))=h(b-(t-a))\\ &\qquad=h(a-(t-b))
=h(a+(t-b))=h(t-T)\end{split}\end{equation*}
for any $t\in\R$, and so $h$ has period $2T$.
\end{proof}

\begin{lemma}\label{MaMi}
If $|h|\le 1$, then
$$ \sup_\R h = -\inf_\R h.$$
\end{lemma}

\begin{proof} Let
$$ m:= \inf_\R h \aand
M:=\sup_\R h.$$
By \eqref{7891}, we have
\begin{equation}\label{rt}
W(m)=W(M).\end{equation}
Thus, by Rolle's Theorem, there exists
$\xi\in ( m, M)$ such that $h'(\xi)=0$.

{F}rom \eqref{UN}, we deduce that $\xi\in \{-1,0,+1\}$.
But since, by assumption, both $m$ and $M$
lie in $[-1,1]$, we have that $\xi\in (m,M)\subseteq
(-1,1)$ and so $\xi=0$.

This says that $m < 0< M$. Thus the claim follows
from \eqref{UN} and \eqref{rt}.
\end{proof}

\begin{lemma}
Suppose that $h$ is either non-periodic or it is constant
but not zero.
Then,
\begin{equation}\label{spe}
W\Big( \inf_\R h\Big)=
W\Big( \sup_\R h\Big)=0.\end{equation}
\end{lemma}

\begin{proof} If $h$ is constant but not zero,
then either $h$ is constantly equal to $-1$
or it is constantly equal to $+1$, because of \eqref{UN}.

Since in such cases \eqref{spe} is obvious,
we focus on the case in which $h$ is not periodic.
Then, by Lemma \ref{2orm},
\begin{equation}\label{1a}
{\mbox{$h$ has at most one
critical point.
}}
\end{equation}
In particular,
$h$ attains either its $\sup$ or its $\inf$ at either $+\infty$
or $-\infty$. So, let us assume, for definiteness
that
\begin{equation}\label{71}
\sup_\R h=
\lim_{t\rightarrow +\infty}h(t),\end{equation}
the other cases being analogous.

By \eqref{17889}, we obtain that the limit
in \eqref{71} holds in $C^1$, therefore
$$ 0= \lim_{t\rightarrow+\infty}
\int_\R h'(s+t) \phi'(s)+
W'(h(s+t))\phi(s)\,  ds=
\int_\R W'\Big( \sup_\R h\Big) \phi(s)\,ds,$$
for any $\phi\in C^\infty_0(\R)$.

This says that
$$ W'\Big( \sup_\R h\Big)=0$$
and so, by \eqref{UN},
$$ \sup_\R h\in \{-1,0,+1\}.$$
If $ \sup_\R h\in \{-1,+1\}$, then
\eqref{spe} holds true, recalling \eqref{7891}.

Thus, we consider the case
$$ \sup_\R h=0.$$
Then, recalling \eqref{1a}, we have two
possibilities: either
\begin{equation}\label{72}
\inf_\R h=
\lim_{t\rightarrow -\infty}h(t),\end{equation}
or there exists $t_o\in\R$ such
that
\begin{equation}\label{73} h(t_o)=\inf_\R h
.\end{equation}
Now, if \eqref{72} holds, we repeat the argument
after \eqref{71} to obtain
that
$$ W'\Big( \inf_\R h\Big)=0$$
and so, by \eqref{UN},
\begin{equation}\label{a7888}
\inf_\R h\in \{-1,0,+1\}.\end{equation}
Since $h$ is not constantly equal to zero,
we have that
$$ 0=\sup_\R h> \inf_\R h$$
and so \eqref{a7888} means that
$\inf_\R h=-1$. This implies
that
\eqref{spe} holds true, recalling \eqref{7891}.

Thus, we have only to deal
with the case in which \eqref{73} holds,
which we now show that is impossible.
Indeed, if \eqref{73}
were true, we would have $h'(t_o)=0$
and so, by \eqref{7891},
$$ W(0)=W\Big(\sup_\R h\Big)=
W(h(t_o)).$$
Thus, by \eqref{UN}, we would have that
$h(t_o)=0$. That is,
$$ \sup_\R h=0=\inf_\R h,$$
in contradiction with our assumptions.
\end{proof}

\begin{lemma}\label{0aoo}
Let $a<b$. If $h$ is monotone in $(a,b)$ then
$$ \int_a^b |h'(t)|\,dt \le 2
\sup_\R |h|.$$
\end{lemma}

\begin{proof} We have\begin{equation*}
\int_a^b |h'(t)|\,dt = \left| \int_a^b h'(t)\,dt\right|=|h(b)-h(a)|\leq
2\sup_\R |h|.\qedhere
\end{equation*}\end{proof}

\begin{lemma}
Suppose $|h|\le 1$. Then, if $h$ is not periodic,
$$ \int_{-\infty}^{+\infty}|h'(t)|\,dt \le 4.$$
\end{lemma}

\begin{proof} By Lemma \ref{2orm}, we see
that only two cases hold: either $h'$ never vanish
or $h'$ has only one zero.

In any case, there exists $c\in \R$ in such a way that
$h'(t)\ne0$ for any $t\in (-\infty, c)\cup(c,+\infty)$.

Consequently, by Lemma \ref{0aoo}, for any $a<c<b$,
$$ \int_a^b|h'(t)|\,dt =\int_a^c |h'(t)|\,dt+
\int_c^b |h'(t)|\,dt\le
2\sup_\R |h|+2\sup_\R |h|.
$$
Then, the desired result follows by sending
$a\rightarrow-\infty$ and
$b\rightarrow+\infty$.
\end{proof}

\begin{lemma}\label{jajjjk}
Let
$$ \sigma \in \left\{ \inf_\R h, \sup_\R h\right\}.$$
Suppose $|h|\le 1$. Then,
\begin{equation}\label{aii88}
\int_{-\infty}^{+\infty} \frac{|h'(t)|^2}{2}+
W(h(t))-W(\sigma) \,dt \le C,\end{equation}
for a suitable structural constant $C>0$,
unless
$h$ is periodic and non-constant.
\end{lemma}

\begin{proof}
Since \eqref{aii88} is obvious for $h$ constant,
we focus on the case in which $h$ is not periodic.

We exploit Lemma \ref{intepri},
\eqref{17889}
and \eqref{7891} to conclude that
\begin{equation*}
\begin{split}
&\!\!\! \int_{-\infty}^{+\infty} \frac{|h'(t)|^2}{2}+
W(h(t))-W(\sigma) \,dt \\
&=
\int_{-\infty}^{+\infty} {|h'(t)|^2} \,dt \\
& \le
C\int_{-\infty}^{+\infty} {|h'(t)|} \,dt
\\ &\le 4C.\qedhere
\end{split}
\end{equation*}
\end{proof}

\subsection{A compactness result}

We now point out a useful compactness criterion:

\begin{lemma}\label{CCC}
Let $u_k$ be a sequence of
solutions of \eqref{main pde} in the whole $\R^2$.
Then, up to subsequence,
$u_k$ converges locally uniformly to some
$u$ which is also a solution of
\eqref{main pde} in the whole $\R^2$.
\end{lemma}

\begin{proof}
By Grushin-elliptic regularity (see, e.g.,
\cite{Franchi, FS, K, VCSC}),
we have that
$$ \| u_k \|_{C^{\alpha} (\R^2) } \leq \bar C$$
for some $\bar C>0$,
therefore
up to subsequence,
$u_k$ converges locally uniformly to some
$u$, and so
\begin{equation}\label{Li2}
{\mbox{$u$ is continuous.}}
\end{equation}
Moreover, for any $a\in(0,1)$, the Grushin operator
$\Delta_G$
is uniformly
elliptic in $D_a:=\{ |x|\in (a,1/a)\}$, therefore
standard elliptic estimates give that
$$ \| u_k \|_{C^{2,\beta_a} (D_a) } \leq \bar C_a$$
and so
\begin{equation}\label{88a880}
\Delta_G u (x)= W'(u(x))
\end{equation}
for any $x\in D_a$.

Since $a$ can be taken arbitrarily small,
we have that
\eqref{88a880}
holds for any $x\in\R^2\setminus\{0\}$.
But then, since the map $x\mapsto W'(u (x))$
is continuous, by means of \eqref{Li2}, it follows that
$\Delta_G u$ is continuous too and so
\eqref{88a880}
holds for any $x\in\R^2$.
\end{proof}

\subsection{Basic spectral theory}

\begin{lemma}\label{SPEC}
Fix $a\in(0,1)$, $R\ge 1$. Let~$x_o=a+R$
and $\Omega:= B_R (x_o,0)$.

Then, there exists~$\lambda\in \R$ for which
there exists a non-trivial solution~$\phi$
of
\begin{equation}\label{EI}
\left\{
\begin{matrix}
\Delta_G \phi+\lambda \phi=0 & {\mbox{ in $\Omega$}},\\
\phi=0 & {\mbox{ on $\partial\Omega$}}.
\end{matrix}\right.
\end{equation}
Moreover, we can take
\begin{equation}\label{EI2}
\lambda \in \Big( 0, \frac{C}{R^2}\Big],
\end{equation}
for a suitable~$C>0$.
\end{lemma}

\begin{proof}
We have
\begin{equation}\label{C0}\begin{split}
&\int_\Omega |\nabla_G v|^2\le
\int_\Omega |\partial_x v|^2+(x_o+R
)^2|\partial_y v|^2
\\&\qquad
\le (1+a+2R)^2 \int_\Omega |\nabla v|^2.
\end{split}\end{equation}
Therefore, by standard
Poincar\'e inequality,
\begin{equation}\label{C1}
\int_\Omega |\nabla_G v|^2\le C(a,R) \int_\Omega |v|^2
,\end{equation}
for any~$v\in C^\infty_0(\Omega)$.

Moreover,
\begin{equation}\label{C2}\begin{split}
&\int_\Omega |\nabla_G v|^2\ge
\int_\Omega |\partial_x v|^2+ a |\partial_y v|^2
\\&\qquad
\ge \min\{1,a\} \int_\Omega |\nabla v|^2.
\end{split}\end{equation}

{F}rom~\eqref{C1}, we thus follow the standard
minimization argument,
taking
\begin{equation}\label{C7}
\lambda\,=\,
\inf_{v\in C^\infty_0(\Omega)\setminus\{0\}}
\frac{
\int_\Omega |\nabla_G v|^2
}{
\int_\Omega v^2
}
\end{equation}
and we
recover compactness
from~\eqref{C2} and the classical embeddings,
proving~\eqref{EI}.

Now, if~$\lambda$ and~$\phi$ satisfy~\eqref{EI},
we may suppose that
$$ \int_\Omega \phi^2 = 1$$
and so
$$ \int_\Omega |\nabla_G \phi|^2 =\lambda $$
which gives that~$\lambda> 0$.

Also, from~\eqref{C7}
and a change of variable,
\begin{eqnarray*}
\lambda&=&
\inf_{v\in C^\infty_0(B_R(x_o,0))\setminus\{0\}}
\frac{
\int_{\{(x-x_o)^4+4y^2\le R^4\}} |\partial_x v(x,y)|^2+|x|^2
|\partial_y v(x,y)|^2
\,d(x,y)
}{
\int_{\{(x-x_o)^4+4y^2\le R^4\}} |v(x,y)|^2\,d(x,y)
}
\\
&=&

\inf_{\psi\in C^\infty_0(B_1 )\setminus\{0\}}
\frac{
\int_{B_1} R^{-2} (\partial_w \psi(w,z))^2
+R^{-4} |x_o+R w|^2 (\partial_z \psi(w,z))^2
}{
\int_{B_1} \psi^2
}\\
&\le&
\inf_{\psi\in C^\infty_0(B_1 )\setminus\{0\}}
\frac{
\int_{B_1} R^{-2} (\partial_w \psi(w,z))^2
+R^{-4} (3R)^2 (\partial_z \psi(w,z))^2
}{
\int_{B_1} \psi^2
}\\ &\le&
10 R^{-2}
\inf_{\psi\in C^\infty_0(B_1 )\setminus\{0\}}
\frac{
\int_{B_1} |\nabla \psi|^2
}{
\int_{B_1} \psi^2
}.
\end{eqnarray*}
This and the classical
Poincar\'e inequality imply~\eqref{EI2}.
\end{proof}

\subsection{Extension of bounded harmonic functions}

\begin{lemma}\label{EXTE}
Let~$u$ be~$\Delta_G$-harmonic in~$B_r\setminus\{0\}$.
Suppose that~$u$ is bounded in~$B_r\setminus\{0\}$.
Then, it may be extended to a $\Delta_G$-harmonic
in~$B_r$.
\end{lemma}

\begin{proof}
The fundamental solution of~$\Delta_G$ is
$\psi(x,y)=(x^4+4y^2)^{\frac{-1}{4}}$
(see  Theorem~3.1 of~\cite{Bieske} for a formula for generalized Grushin operators).
Thus, the argument on pages 16--17
of~\cite{LIN} may be repeated verbatim.
\end{proof}

\section{Monotonicity and stability}\label{S0}

We show that \eqref{Mo} is sufficient
for stability. This is in analogy with
the fact that monotonicity in any direction
implies stability in the Euclidean setting
(see \cite{AAC}) -- but in the Grushin plane
the directions do not play the same role, thus
\eqref{Mo} somehow selects the good direction for
stability.

\begin{proposition}\label{677888188}
Let $u\in C^2(\R^2)$ be a solution
of \eqref{main pde} satisfying \eqref{Mo}.

Then, $u$ is stable.
\end{proposition}

\begin{proof} The argument we present here is
a modification of a classical one (see \cite{AAC}
and also Section~7 in~\cite{FSV}
for a general result).
We recall that we need to prove that for any smooth $\phi$, compactly supported
$$0\le\quad \int_{\R^n} |\nabla_G \phi|^2-f'(u) \phi^2\,dx.$$

For any $\varphi$ smooth and compactly supported, we have
\begin{eqnarray*}
&& \int_{\R^2} f'(u) Tu \varphi
=\int_{\R^2} \partial_y \big(f(u)\big) \varphi
=-\int_{\R^2} f(u)\partial_y \varphi
\\&& \quad
\int_{\R^2} \Delta_G u \partial_y \varphi
=
-\int_{\R^2} <\nabla_G u, \nabla_G \partial_y \varphi>
\\&& \qquad
=\int_{\R^2} <T \nabla_G u, \nabla_G \varphi>
=
\int_{\R^2} <\nabla_G (Tu), \nabla_G \varphi>.
\end{eqnarray*}

Therefore, by taking $\varphi:=\phi^2/ (Tu)$,
and
making use of the Cauchy-Schwarz inequality,
\begin{equation*}\begin{split}
& 0  \int_{\R^n}\frac{ 2\phi < \nabla_G (Tu), \nabla_G
\phi > }{Tu}-
\frac{\phi^2 |\nabla_G (Tu)|^2
}{(Tu)^2}-f'(u)\phi^2\,dx\\
&\quad\quad \le
\int_{\R^n} |\nabla_G \phi|^2-f'(u) \phi^2\,dx.
\qedhere\end{split}
\end{equation*}\end{proof}

\section{Energy estimates}\label{S2}

We follow here some ideas of \cite{AAC}
to estimate the energy
$$\cF_R (u):= \int_{B_R(0)}
\frac{|\nabla_G u(\xi)|^2}{2}+W(u(\xi))\,d\xi.$$

For this, for any $t\in\R$, we define the translation
$$ u^t (x,y):= u(x,y+t)$$
and the translated energy
$$ \cE_R (t):= \cF_R (u^t).$$
Of course, $\cE_R(0)=\cF_R(u)$.

\begin{lemma}
Suppose that $u\in C^2(\R^2, [-1,1])$, with $Tu>0$
and $|\nabla_G u|\in
L^\infty(\R^2)$,
is a solution of
$$ \Delta_G u(\xi)=W'(u(\xi))
\qquad{\mbox{ for any $\xi\in\R^2$.}}$$
Then, there exists a structural constant $C$
in such a way that
\begin{equation}
\label{enes}
\cE_R (0) \leq \cE_R(t)+CR^2,\end{equation}
for any $t\in\R$ and any $R>0$.
\end{lemma}

\begin{proof}
We prove \eqref{enes} for $t>0$ (this is enough, since
$u(x,-y)$ is also a solution).

We have, recalling Lemma \ref{Green},
\begin{eqnarray*}
\frac{d}{dt} \cE_R (t)&=&
\int_{B_R(0)} < \nabla_G u^t ,\nabla_G T u^t>
+W'( u^t) T u^t\, d\xi \\
&\ge& \int_{B_R(0)} \Big( - \Delta_G ( T u^t)
+W'( u^t) \Big) T u^t \,d\xi\\&&
-R^2
\|\nabla_G u^t\|_{L^\infty(\R^2)}
\int_{\partial B_1(0)}
Tu^t (RX,R^2Y)
\,d\cH^1(X,Y)
\\ &=& 0
-R^2
\|\nabla_G u \|_{L^\infty(\R^2)}
\int_{\partial B_1(0)}
\partial_y u(RX, R^2Y+t)
\,d\cH^1(X,Y).
\end{eqnarray*}
Hence, fixed any $\tau>0$,
\begin{equation*}\begin{split}
\cE_R (\tau)- \cE_R (0)&=
\int_0^\tau \frac{d}{dt} \cE_R (t) \,dt
\\ &\ge -R^2 \| \nabla_G u\|_{L^\infty(\R^2)}
\int_0^\tau
\int_{\partial B_1(0)} \partial_y u(
RX,R^2Y+t)
\,d\cH^1(X,Y)\,dt\\&=
-R^2
\| \nabla_G u\|_{L^\infty(\R^2)}
\int_{\partial B_1(0)} \int_0^\tau \partial_y u(RX,
R^2Y+t)
\,dt\,d\cH^1(X,Y)
\\&= -R^2
\| \nabla_G u\|_{L^\infty(\R^2)}
\int_{\partial B_1(0)} u(RX,R^2Y+\tau)-u(RX,R^2Y)
\,d\cH^1(X,Y)\\ &\ge -2R^2
\| \nabla_G u\|_{L^\infty(\R^2)}
\| u\|_{L^\infty(\R^2)}\, \cH^1(\partial B_1(0)),
\end{split}\end{equation*}
which gives \eqref{enes}.\end{proof}

\begin{theorem}\label{EB}
Suppose that $u\in C^2(
\R^2, [-1,1])$, with $|\nabla_G u|\in
L^\infty(\R^2)$
is a solution of
$$ \Delta_G u(\xi)=W'(u(\xi))
\qquad{\mbox{ for any $\xi\in\R^2$.}}$$
Assume that \eqref{Mo} holds true.

Then, there exists a structural constant $C$
in such a way that
$$ \cF_R (u) \le C R^2$$
for any $R>0$.

As a consequence, \eqref{enesty}
holds true.
\end{theorem}

\begin{proof} We have that $u$ is bounded and monotone
in $y$, thanks to \eqref{Mo}.
Thus, we may define
\begin{equation}\label{Li}
u^\pm (x):=\lim_{y\rightarrow\pm \infty} u(x,y).
\end{equation}
Then, from Lemma \ref{CCC}, we have that
\begin{equation}\label{88a8801}
\Delta_G u^\pm (x)= W'(u^\pm (x))
\end{equation}
In fact, since $u$ does not depend on $y$,
we may write \eqref{88a8801} as
\begin{equation}\label{35bis}
(u^\pm)''(x)=W'(u^\pm(x))\end{equation}
and so we may apply to $u^\pm$
the ODE analysis developed in Section \ref{odes}.

For this, we observe that
\begin{equation}\label{noncp}
{\mbox{at least one between $u^+$ and $u^-$
is either
constant or non-periodic.}}
\end{equation}
To prove \eqref{noncp}, we argue by contradiction,
supposing that $u^+$ and $u^-$ are both
periodic and non-constant. In particular,
by Cauchy Uniqueness Theorem, $|u^\pm|<1$ and
then, by Lemma \ref{MaMi}, we would have that
\begin{equation}\label{bi1}
\max_\R u^\pm  = -\min_\R u^\pm.\end{equation}
But from \eqref{Mo}, we know that
\begin{equation}\label{bi2}
{\mbox{$u^+(x)>u^-(x)$
for any $x\in\R$}}\end{equation}
and so, if we set $x^\pm_{\min}$,
$x^\pm_{\max}$ be such that
$$
u^\pm(x^\pm_{\min})=\min_\R u^\pm
\aand
u^\pm(x^\pm_{\max})=\max_\R u^\pm
,$$
we deduce from \eqref{bi1} and \eqref{bi2}
that
\begin{eqnarray*}
&& u^- (x^+_{\min}) \ge u^- (x^-_{\min}) = - u^-
(x^-_{\max})
> -u^+ (x^-_{\max})\\
&& \qquad
\geq -u^+(x^+_{\max})
= u^+(x^+_{\min}) >u^- (x^+_{\min}).
\end{eqnarray*}
This contradiction proves \eqref{noncp}.

We now claim that
\begin{equation}\label{noncpsta}
{\mbox{either $u^+$ or $u^-$ is non-periodic
or constant but not zero.}}
\end{equation}
To prove this, we argue by contradiction.
Suppose \eqref{noncpsta} is false. Then,
both $u^-$ and $u^+$ are periodic.
Then, at least one, say $u^+$ is constant,
because of \eqref{noncp}.
If $u^+$ were not equal to zero, then \eqref{noncpsta}
would be true, thus we have to say that $u^+$
is constantly equal to zero
and that $u^-$ is periodic. But then $u^-$
cannot be constant, otherwise
\eqref{35bis}, \eqref{UN} and \eqref{Mo}
would say that $u^-$ is constantly equal to $-1$
and \eqref{noncpsta} would be true.
Thence, we are forced to the case in which $u^+$
is identically zero and
$u^-$ is periodic and non-constant.
Thus, by \eqref{Mo},
$$ \sup_\R u^- \le 0$$
and so, by
Lemma \ref{MaMi},
$$ \inf_\R u^- =-\sup_\R u^- \ge 0\ge
\sup_\R u^-.$$
This would say that $u^-$ is constant,
while we know it is not the case.

This contradiction proves \eqref{noncpsta}.

By means of \eqref{noncpsta}, up to a sign change, we
may suppose that
$u^+$ is either constant but not zero or it
is non-periodic.
Consequently, by
Lemma \ref{jajjjk},
\begin{equation}\label{akk1}
\int_{-\infty}^{+\infty} \frac{|(u^+)'(t)|^2}{2}+
W(u^+(t))-W(\sigma^+) \,dt \le C^+,\end{equation}
for a suitable $C^+ >0$, with
$$ \sigma^+ \in \Big\{ \inf_\R u^+,\sup_\R u^+
\Big\}.$$
In fact, \eqref{spe}
and \eqref{akk1} give that
\begin{equation}\label{akk2}
\int_{-\infty}^{+\infty} \frac{|(u^+)'(t)|^2}{2}+
W(u^+(t))\,dt \le C^+.\end{equation}

Moreover, by \eqref{enes},
\begin{eqnarray*}
\cE_R (0) -CR^2
&\leq& \lim_{t\rightarrow+\infty}\cE_R(t)
\\&=&\lim_{t\rightarrow+\infty}
\int_{B_R(0)}
\frac{|\nabla_G u(x,y+t)|^2}{2}+W(u(x,y+t))\,d(x,y)
\\ &=&
\int_{B_R(0)}
\frac{|\nabla_G u^+ (x)|^2}{2}+W(u^+(x))\,d(x,y)
\\ &\leq&
\int_{ -R^2/2}^{R^2/2}
\int_{ -\infty}^{ +\infty}
\frac{|\nabla_G u^+ (x)|^2}{2}+W(u^+(x))\,dx\, dy
\\ &=&
R^2
\int_{ -\infty}^{ +\infty}
\frac{|\nabla_G u^+ (x)|^2}{2}+W(u^+(x))\,dx
.\end{eqnarray*}
Thus, by \eqref{akk2},
\begin{equation*}
\cE_R (0) -CR^2
\le C^+R^2.
\qedhere\end{equation*}
\end{proof}

\bigskip

\section{The counter-example}\label{S3}
\subsection{Monotonicity and Maximum Principle}

For any $s\in\R$ and $\xi\in\R^2$,
let $$T_s\xi:=\xi+(0,s).$$
A domain $\Omega \subset \R^2$ is said to be $T$-convex
if for any
$\xi_1 \in \Omega$ and any $ \alpha> 0 $ such that  $ T_\alpha \xi_1 \in \Omega$  one has that
$T_s\xi_1 \in \Omega$ for every $s \in (0,\alpha)$.

That is, $\Omega$ is $T$-convex when vertical segments joining
two points of $\Omega$ lie in $\Omega$.

\begin{theorem}\label{mono}
Let $\Omega$
be an arbitrary bounded domain of $\R^2$ which is
$T$-convex.

Let $ u \in \Lambda^2 (\Omega) \cap C(\bar{\Omega})$
be a solution of
\begin{equation}\label{main}
\left. \begin{array}{rll}
\Delta_G u + f(u) & = & 0 {\rm ~~ in ~~} \Omega\\
        u & = & \psi{\rm ~~ on~~} \partial \Omega
\end{array}\right\}
\end{equation}
where $f$ is a Lipschitz continuous function. Assume that for any
$\xi_1$, $\xi_2 \in \partial \Omega$, such that $ \xi_2 = T_\alpha \xi_1$ for some $\alpha> 0$,  we have, for each $s \in (0, \alpha)$ either
\begin{equation}\label{1}
\psi(\xi_1) < u(T_s\xi_1) <
\psi(\xi_2){\rm~~ if ~~} T_s\xi_1\in \Omega
\end{equation}
or
\begin{equation}\label{2}
\psi(\xi_1) < \psi(T_s\xi_1) < \psi(\xi_2){\rm~~ if ~~}
   T_s\xi_1\in  \partial\Omega.
\end{equation}

Then $u$ satisfies

\begin{equation}{\label{result}}
u (T_{s_1}\xi) < u (T_{s_2}\xi)
 \end{equation}
for any $0<s_1 < s_2<\alpha$ and  for every $\xi \in \Omega$.

Moreover, $u$ is the unique solution of (\ref{main})
 in  $\Lambda^2(\Omega) \cap C(\bar{\Omega})$ satisfying (\ref{1}).
\end{theorem}

The proof of this
result
is done through the sliding method
introduced
in \cite{BN} for uniformly elliptic equations.
This method uses two fundamental ingredients:
the Maximum Principle in small domains and
the invariance of the operator
with respect to ``sliding". In \cite{BP} the
equivalent of Theorem \ref{mono} was proved for
sub-elliptic equations in nilpotent Lie groups.
There, the key ``new" ingredient being a H\"older estimate
for H\"ormander type operators proved
in \cite{K} that
allowed to prove the Maximum Principle in small domains.

The operator is invariant by $T_s$
translations
and our equation satisfies the hypotheses of
\cite{K}, hence the proof
of Theorem \ref{mono}
proceeds exactly like the one given in \cite{BP},
and we omit it.

\subsection{Existence of monotone solutions
that are not one-di\-men\-sion\-al}

The following result shows that Question \ref{QQQ}
has a negative answer:

\begin{theorem}\label{MMM}
There exists a solution of
$$ \Delta_G u-W'(u)=0$$
in $\R^2$ such that $Tu=\partial_y u>0$.

Also, such $u$ is not one-dimensional.
\end{theorem}

\begin{proof}
We follow the two
steps of \cite{BL}.

{\bf Step 1.} {\it Construction of a monotone solution in a
bounded set.}

Let $M>0$ be greater then the Lipschitz constant
of $f$, let $g(u):=f(u)+Mu$, $Q_R^+:=(-R,R)
\times[0,R^2]$ and~$Q_R^-:=(-R,R)\times(-R^2,0)$.

We consider the operator ${\cal T}$
on~$C^{\alpha}$ such that~${\cal
T}v=u$ is the classical solution of
$$\left\{\begin{array}{lc}
\Delta_G u -Mu=-g(v) & \mbox{in }\ Q_R^+\\
u(x,0)=0,\ u(x,R^2)=1, & u(-R,y)=\psi(y),\ u(R,y)=\psi(y)
\end{array}\right.
$$
where $0\leq \psi\leq 1$ with $\psi(0)=0$ and $\psi(R^2)=1$.

The following properties hold:

(P1) ${\cal T}$ is well defined, see \cite{VCSC,K}.

(P2) It is monotone, i.e.  $0\leq v_1\leq v_2 \leq 1$ implies
${\cal T}v_1\leq {\cal T} v_2$.
This is just the Maximum Principle,
because with our choice of $M$ we get that
$$v_1\leq v_2\Longrightarrow g(v_1)\geq g(v_2).$$

(P3) If $0\leq v\leq 1$ then $0\leq {\cal T}
v \leq 1$ (again by
Maximum Principle).

(P4) For $R$ sufficiently large there
exists $v_o>0$ in some fixed subset of $Q_R^+$ such that
if $u_k:={\cal T}^k(v_o)$ then $u_k\geq v_o$ for any $k\in N$.

Let
us prove~(P4).

Let $$l:
=\lim_{s\rightarrow 0}\frac{|W'(s)|}{s}$$ and let $R_o$
be
sufficiently large that $Q_{R_o}^+$  contains a
ball $B$ such that  $B\cap \{ x=0\}=\emptyset$ and
$\lambda_o$ the principal eigenvalue of $-\Delta_G$ in $B$
satisfies
$$\lambda_o\leq \frac{l}{2}.$$
We remark that we can take such a~$\lambda_o$
in the light of Lemma~\ref{SPEC}.

Let $\varphi_o$ be the corresponding
eigenfunction normalized by $\sup \varphi_o=1$.
Our choice of $\lambda_o$ implies that there exists $\varepsilon
>0$ such that
\begin{equation}\label{lam}
\lambda_o\varepsilon\varphi_o\leq |W'(\varepsilon\varphi_o)|.
\end{equation}

Now we define
\begin{equation}\label{5.5bis}
v_o=\left\{\begin{array}{ll}
\varepsilon\varphi_o & \mbox{ in }\quad B\\
0 & \mbox{ in }\quad Q_{R}^+\setminus B.
\end{array}
\right.
\end{equation}
Observe that, in $B$, the Grushin
operator
$\Delta_G$ is uniformly elliptic, and so
by standard estimates we know that $v\in C^\alpha(Q_R^+)$.
Using (\ref{lam}),  (P2) and (P3),  we get that
$u_o={\cal T}(v_o)\geq v_o$.

So, iteratively, ${\cal T}^k(v_o)\geq v_o$
for any~$k\in\N$.
This proves (P4).

By Lemma \ref{CCC}, we may and do
suppose that $u_k
:={\cal T}^k(v_o)
$ converges to a solution $\tilde u=\tilde u_R$ of
$$\left\{\begin{array}{lc}
\Delta_G \tilde u -W'(\tilde u) =0 & \mbox{in }\ Q_R^+\\
\tilde u(x,0)=0,\
\tilde u(x,R^2)=1 & \tilde u(-R,y)=\psi(y),\
\tilde u(R,y)=\psi(y)
\end{array}\right.
$$
Note that
$\tilde u$ satisfies $0\leq \tilde u
\leq 1$. Hence, using Theorem \ref{mono}, we
know that \begin{equation}\label{8a88kkkk1}
\partial_y \tilde u>0.\end{equation}

Finally, we extend the solution to $Q_R=\overline{Q_R^+}\cup \overline{Q_R^-}$ by taking
$$v_R(x,y):=\left\{\begin{array}{lc}
\tilde u(x,y) & \mbox{for} \ (x,y)\in \overline{Q_R^+}\\
-\tilde u(x,-y)& \mbox{for} \ (x,y)\in\overline{Q_R^-}
\end{array}
\right.
$$
Clearly, $v_R$ is a solution in
$Q_R^+\cup {Q_R^-}$. Also the solution $u$ is $C^2$ up to the boundary for $x\neq 0$. Hence we get that $v_R$ is a solution in $Q_R\setminus\{(0,0)\}$.
To check that $v_R$ is a solution in all of $Q_R$. Observe that
the map $\zeta\mapsto
W'(v_R(\zeta))$ is in $C^\alpha(Q_R)$, hence there
exists $w\in C^{2,\alpha}_{loc}(Q_R)
\cap C(\overline{Q_R})$
solution of
$$\Delta_G w-W'(v_R)=0 \quad \mbox{in}\quad Q_R.$$
Then $w-v_R$ is $\Delta_G$-harmonic in
$Q_R\setminus\{(0,0)\}$ and
it is bounded. Thus, by Lemma~\ref{EXTE},
it is $\Delta_G$-harmonic in
$Q_R$,
and so~$v_R$ is a solution in all of $Q_R$.

Furthermore $v_R$ is monotone in $T$,
in the sense that $Tv_R=\partial_y v_R>0$, because
of \eqref{8a88kkkk1}.

{\bf Step 2.} {\it Let $R\rightarrow \infty$.}
Then,
by Lemma \ref{CCC}, $v_R$ locally uniformly
converges to some $u$, which is
a solution of
$$\Delta_G u-W'(u)=0 \quad \mbox{in}\quad \R^2.$$
Furthermore, in $Q^+_R$,
$$ v_R=\tilde u=\lim_{k\rightarrow+\infty}
u_k\ge v_0,$$
due to~(P4)
and so, by~\eqref{5.5bis},
$u\not\equiv 0$
in $Q_R^+$.

Then, $u$ is monotone i.e. $\partial_y u>0$, and
it is therefore the counter-example we are looking for.

Indeed, $u$ is not one-dimensional;
suppose, by contradiction,
that there exists a function $g$ such that
$$u(x,y)
=g(ax+by),$$
for any $(x,y)\in\R^2$.

Then, the strict monotonicity
in $T$ of $u$ implies that
\begin{equation}\label{b0}
b\neq 0.\end{equation}

Clearly $g$ would be a solution of
\begin{equation}
\label{67}
(a^2+b^2x^2)g''(ax+by)-W'(g(ax+by))=0,
\end{equation}
for any $(x,y)\in\R^2$.

This implies that for any $t$ along the lines $ax+by=t$,
$$(a^2+b^2x^2)g''(t)-W'(g(t))=0.
$$
Since $b\neq 0$, this implies that $g''\equiv 0$. Hence $W'(g(t))= 0$
for any $t$
and so $g$ would be constant, in
contradiction with the fact that $Tu>0$.
\end{proof}

\section*{Acknowledgements}
Partially supported by MIUR Metodi variazionali
ed equazioni differenziali non-lineari
and by MIUR Metodi di viscosit\`a, metrici
e di teoria del controllo
in equazioni alle derivate parziali
non-lineari.


TITLE: Word Problem (Circles/Rectangles)
QUESTION [0 upvotes]: Question: The  diagram shows a $400m$ running track. The "Bends" (semi cricles) are $100m$ and the "straights" are $100m$. Find the distance run by an athlete who runs $x$ metres out from the kerb. (Had to make the diagram in paint because the picture is not really clear so apologies)

The part where I am stuck is what it means by runs $x$ metres out from the kerb.
The answer is $400 + 2\pi x$ but not too sure how they got it

REPLY [1 votes]: I'm not sure if you're picture is "wrong" or just "the best you could do".  The picture should look like the following:

Assuming that this is the "interior" then the radius of the two semi-circles is ($180^\circ = \pi$):
$$
\pi r = 100 \rightarrow r = \frac{100}{\pi}
$$
Then adding to $x$ meters to the radius would give:
\begin{align}
d =&\ 200 + 2(r + x)\pi \\
=&\ 200 + 2 r\pi  +2x\pi \\
=&\ 200 + 2\cdot 100 + 2x\pi \\
=&\ 400 + 2x\pi
\end{align}
This is using the fact that $r\pi=100$ because the two semi-circle loops are $100$ meters.


text:
\begin{document}
\maketitle
\beg{abstract}We study the angles between the eigenvectors of a random $n\times n$ complex matrix $M$ with density $\propto \mathrm{e}^{-n\operatorname{Tr}V(M^*M)}$ and $x\mapsto V(x^2)$ convex. We prove that for unit eigenvectors $\mathbf{v},\mathbf{v}'$ associated with distinct eigenvalues $\lambda,\lambda'$ that are the closest to specified points $z,z'$ in the complex plane, the rescaled inner product $$\sqrt{n}(\lambda'-\lambda)\langle\mathbf{v},\mathbf{v}'\rangle$$ is uniformly sub-Gaussian, and give a more precise statement in the case of the Ginibre ensemble.\en{abstract}

 \section{Introduction and main results}

 \subsection{Setup and main results}
 Let $V:\R^+\to \R$ be a function \st the following holds.
\begin{equation}
  \label{eq-alpha}
  \mbox{\rm For some $\al>0$, the function $V(x^2)-\f\al2{x^2}$ is convex.}
\end{equation}
 Let $X$ be an $n\ti n$ complex matrix with law
 \begin{equation}
   \label{eq-law}
   \propto \me^{-n\Tr V(M^*M)}\ud M,
 \end{equation}
 where
 $\ud M$ is the standard Lebesgue measure on $n\ti n$ complex matrices. In particular,
 all eigenvalues of $X$ are distinct, almost surely (see \rem{whyeigenvalyesaredistinct}).

 Let $z,z'\in \C$ and let $\lam$ and $\lam'$ denote the eigenvalues of $X$ that are the closest to respectively  $z$ and $z'$ (if $z=z'$ or $\lam$ is the closest eigenvalue to both $z$ and $z'$, then $\lam'$
 is the second closest to $z'$).
 Let  $\bv$ and $\bv'$ denote some associated eigenvectors of unit $\ell_2$ norm.  We want to study the quantity $n|\lam'-\lam||\lan \bv,\bv'\ran|^2$, which leads us to
introduce the random variable $Y$, defined through any of the two following equivalent equations  \be\la{def:Yt12} Y\;:=\;  n \f{|\lam'-\lam|^2}{|\lan\bv,\bv'\ran|^{-2}-1}\ee  or
\be\la{def:Yt12_2} n|\lam'-\lam|^2 |\lan\bv,\bv'\ran|^2\;=\;\f{ Y}{ \f{Y}{n|\lam'-\lam|^2}+1}, \qquad Y>0\ee
Since eigenvalues are almost surely distinct and
$|\lan \bv, \bv'\ran|$ is invariant under multiplication of the eigenvectors by
a complex scalar of norm $1$,  the random variable $Y$ is well defined.

Recall $\alpha$
from \eqref{eq-alpha}.
Our first main result is the following.

\beg{Th}\la{thCVeigvectors}The random variable $Y$ satisfies

\be\la{eqYt12}\p(Y\ge\del)\;\le\;2 \exp\lf(-\f{\al}2\del\ri), \quad
\mbox{\rm for any $\del>0$} .\ee
\en{Th}
In particular,
when $n|\lam'-\lam|^2\gg 1$, \eqre{def:Yt12_2} shows that
$$  |\lan\bv,\bv'\ran|\approx \f{\sqrt Y}{\sqrt n|\lam'-\lam|},$$
and it follows from Theorem \ref{thCVeigvectors} and a union bound that
all eigenvectors corresponding to mesoscopically separated
eigenvalues are asymptotically orthogonal to each other.

\begin{rmk}\label{rem-exponential}
In the case of the Ginibre ensemble ($V(x)=x$, so that the entries of
$X$ are i.i.d. standard complex Gaussian variables with variance $n^{-1}$),
the random variable
$Y$ has an exponential law of mean $1$. This fact is probably well known,
and
follows from Equations  \eqre{eq:fromYtot12}  and
\eqre{densityTproof}-\eqre{def:ttilde} below. In particular,
$\sqrt{Y}$ is distributed like the norm of a standard complex Gaussian variable.
\end{rmk}
(We recall that  a \emph{standard complex Gaussian variable} is a
centered complex Gaussian random variable  $Z$ \st $\E Z^2=0$ and $\E |Z|^2=1$.)

Our second main result is concerned with Ginibre matrices, for which

we extend an asymptotic version of Remark
\ref{rem-exponential} to the multivariate framework.

\beg{Th}\la{thCVeigvectorsGinibre}Suppose that $V(x)=x$. For a  fixed $k\ge 2$,  let $z_1, \ld, z_k$ be (deterministic) points
in the unit disk, possibly dependent on $n$,
\st for a certain $\eps>0$, uniformly in $n$, \be\la{300518111}  \sqrt n\min_{1\le i<j\le k} |z_j-z_i|\ge n^\eps .\ee
For each $i$, let $\lam_i$ be the eigenvalue that is the closest to $z_i$ and
let $\bv_i$ be
an associated eigenvector. Let $\theta_i$, $i=1,\ldots,k$, be i.i.d.
variables uniformly distributed on the $[0,2\pi]$, independent of $X$.

Then the distribution of the     triangular array
$$\lf(\sqrt n(\lam_j-\lam_i) \lan \me^{\ii\theta_i} \bv_i,
\me^{\ii\theta_j}\bv_j\ran\ri)_{1\le i<j\le k}$$
converges, as $n\to\infty$,  to the distribution of a triangular array of independent standard complex Gaussian variables.
\en{Th}

 \brem The typical distance between two eigenvalues of $X$ that are ``neighbors" of each other in the spectrum of $X$ has order $n^{-1/2}$. Hence, because of Hypothesis \eqre{300518111}, this result is well adapted for most pairs of eigenvalues, but not for those that are as close as possible  (in our proof, Hypothesis \eqre{300518111} is necessary for  estimates \eqre{eq:maxinv1805} to \eqre{300518c} to hold). For one given   pair of eigenvalues at distance $\asymp n^{-1/2}$, much information is contained in the fact that the random variable $Y$ introduced in \eqre{def:Yt12} and \eqre{def:Yt12_2} has exponential distribution with mean one. If one considers not only one such pair, but an arbitrary finite number $k$ of eigenvalues that are at distances of order $n^{-1/2}$, the problem is less simple.
 \erem

 \subsection{Background} The study of eigenvectors of random Ginibre matrices
 seems to have been initiated in \cite{CM1}. For a matrix $X$,
 let $v_i$ (respectively, $w_i$) denote the left (respectively, right)
eigenvectors
corresponding to eigenvalues $\lambda_i$, where
the normalization
$\lan \bv_i,\bw_j\ran=\delta_{ij}$ is imposed. Using the Schur representation
 $X=UTU^*$ with $T$ upper triangular and $U$ unitary,
 they
 computed, for the Ginibre ensemble,
 the correlations of eigenvectors and cross correlations of right
 and left eigenvectors, with special emphasis on the correlator
 \begin{equation}
   \label{eq-cor}
   O_{1,2}=\E (\lan \bv_1,\bv_2\ran
\lan \bw_1,\bw_2\ran).
\end{equation}
 Using the joint density of entries of $T$, the
 evaluation of the latter correlations reduce to the evaluation of certain
 Green functions. This point of view was recently significantly expanded to more general models in \cite{NowakTarnowski} (using diagrammatic methods), as well as
 in \cite{CR}, where multi-points correlations are evaluated and related to
 two point correlations. We refer the reader to the introduction of
 \cite{CR} for further details and an extensive bibliography.
Recent works \cite{Fyo,BD} study distributional limits for condition numbers, as well as refined estimates for overlaps in the microscopic and mesoscopic regime.

Another very relevant recent work  is \cite{BNST}, which deals with matrices
$X$ with joint density of entries of the form \eqref{eq-law} (whithout
assuming the convexity of $V$). In this general setup,
the correlator $O_{12}$ from \eqref{eq-cor}
is computed.

 Our results, as well as \cite{CM,CR},
 build
upon the evaluation of the joint distribution of the entries of $T$, see
\cite{Ginibre, TAO2, Mehta}; these derivations
do not address explicitely the ordering of the diagonal
elements in $T$; in our approach, we choose the
ordering as function of the full set of diagonal elements.
For this reason, we provide explicitly a proof
of the joint distribution of entries.

Finally, we mention that general delocalization results for eigenvectors
of random non-Hermitian matrices with independent entries
appear in \cite{RV}.

 \subsection{A conjecture}
The inner products appearing in Theorem \ref{thCVeigvectorsGinibre}
can be written in terms of the off diagonal
entries of the upper triangular matrix $T$
in the Schur decomposition of $X$, see the proof of Theorem
\ref{thCVeigvectors} below. In the Ginibre case, these
entries (scaled by $\sqrt{n}$)
are iid standard complex Gaussians, and for general $V$,
it is still the case that
$Y=n|t_{12}|^2$, see \eqref{eq:fromYtot12} below. This leads us to the following.
\begin{conj}
  Under the assumptions of Theorem
  \ref{thCVeigvectors},  the sequence
  $Y/\E Y$ converges in distribution,
  as $n\to\infty$, to the exponential law of parameter one.
  \end{conj}
 Some preliminary computations make the conjecture plausible. In addition,
 the simulations in
 Figure \re{Fig:Spectral_Resolution} are in agreement with the conjecture.

\begin{figure}[!ht]
\centering
\subfigure[$V(x)=x$ (Ginibre)]{
\includegraphics[scale=.35]{OverlapsGinibre_15040}} \qquad
\subfigure[$V(x)=x+\f{x^4}4+\f{x^5}5$]
{\includegraphics[scale=.35]{OverlapsSRT_150_1_1000_40_1_0_0_1_1}}
\caption{{\bf Universality of eigenvector angles:}  density   $\ds 2x\me^{-x^2}$ (in red) vs the histograms (in blue) of   $\ds   \f{\sqrt n|\lam'-\lam|}{\sqrt{|\lan\bv,\bv'\ran|^{-2}-1}}$ for $\lam\ne  \lam'$ running through the spectrum of a matrix with   distribution $\propto \me^{-n\Tr V(M^*M)}$ ($n=150$, sample of size $40$, sampled thanks to  Langevin Monte Carlo), with 2 different choices for $V$.  The sample on the right has been rescaled so that its empirical second moment is $1$ (because the distribution of $  \f{|\lam'-\lam|}{\sqrt{|\lan\bv,\bv'\ran|^{-2}-1}}$ can only be universal up to a rescaling of the matrix).
}\la{Fig:Spectral_Resolution}
\end{figure}

 \section{Proofs}
 In the proofs below, we use the joint distributions derived in
 Theorem \ref{thEigenvectors} from the appendix.
 \bpr[Proof of \theo{thCVeigvectors}]  Set $\cO:=\cO_{z,z',z',z',\ld, z'}$ (see \eqre{defO} for the definition of this   set).  By Theorem \re{thEigenvectors}, we know that $X$ can be written $X=UTU^*$ with $U$ unitary and $T=[t_{ij}]$ upper triangular having the density
 \be\la{densityT} \propto\one_{(t_{11},\ld, t_{nn})\in \cO }|\Del(t_{11},\ld,t_{nn})|^2 \me^{-n\Tr V(T^*T)},\ee
Hence by definition of
$\cO$, $\lam$ and $\lam'$ are the two first diagonal entries of $T$.

Thus the vectors $\bw:=(1, 0, \ld, 0)$ and $\bw':=\lf(|t_{12}|^2+|\lam'-\lam|^2\ri)^{-1/2}(t_{12}, \lam'-\lam, 0, \ld, 0)$
 are  unit right eigenvectors of $T$ for the eigenvalues $\lam$ and $\lam'$.
Hence as $U$ is unitary, \be\la{CVproblamz0}|\lan\bv,\bv'\ran|=|\lan\bw,\bw'\ran|=\f{|t_{12}|}{\lf(|t_{12}|^2+|\lam'-\lam|^2\ri)^{1/2}},\ee so that
\be\la{eq:fromYtot12} Y=   n \f{|\lam'-\lam|^2}{{|\lan\bv,\bv'\ran|^{-2}-1}}=  n |t_{12}|^2\ee
Thus we have to prove that   for any $\del>0$,  \be\la{eqYt12proof0}\p(\sqrt n|t_{12}|\ge\del)\;\le\;2\me^{-\f{\al\del^2}2}  .\ee

Notice first that for any fixed  $(t_{11}, \ld, t_{nn})\in \cO$, the distribution $\mu_{\cdote|(t_{11}, \ld, t_{nn})}$  of $(t_{ij})_{1\le i<j\le n}$  conditional to $(t_{11}, \ld, t_{nn})$ has  on $\C^{n(n-1)/2}$ a density  \be\la{densityTconditionaldiag}\propto \me^{-n\Tr V(T^*T)}\ee for $T$ the upper-triangular matrix with upper-triangular entries $(t_{ij})_{1\le i\le j\le n}$. Thus by \lemm{lem:2505182} and \rem{rem:2505182}, $\mu_{\cdote|(t_{11}, \ld, t_{nn})}$ satisfies a LSI with constant $(\al n)^{-1}$.
Note also  that for any fixed $\tta_1, \ld,\tta_n\in \R$, the density of $\mu_{\cdote|(t_{11}, \ld, t_{nn})}$ given at \eqre{densityTconditionaldiag} is invariant under the transformation $$(t_{ij})_{1\le i<j\le n}\longmapsto (\me^{\ii(\tta_i-\tta_j)}t_{ij})_{1\le i<j\le n}.$$ We deduce that the expectation of $t_{12}$ with respect to $\mu_{\cdote|(t_{11}, \ld, t_{nn})}$ vanishes,
so that by the LSI,   for any $\del>0$,  \be\la{eqYt12proof}\mu_{\cdote|(t_{11}, \ld, t_{nn})}(\sqrt n|t_{12}|\ge\del)\;\le\;2\me^{-\f{\al\del^2}2}  .\ee Integrating over $(t_{11}, \ld, t_{nn})$, we get \eqre{eqYt12proof0}.
 \epr

 Before the proof of \theo{thCVeigvectorsGinibre}, we
 prove two preliminary lemmas. We suppose here that $V(x)=x$.

  \blem\la{lem:spacing_eigprelim} Let $z_0$ in the unit disk, possibly depending on $n$ and $s\in (0, 1/2)$. Then with \pro tending to one  as $n\to\infty$, $X$ has at least $\f{n^{1-2s}}5$ eigenvalues at distance $\le n^{-s}$ from $z_0$.
  \elem

  \bpr   Let $N_s$ denote the number of eigenvalues in the disk $D(z_0, n^{-s})$.  Let $f$ be a smooth non negative function    with value $1$ on the   disk $D(0,1/2)$ and with support contained in the disk $D(0,1)$.
  Then we have $$N_s\ge \sum_{j=1}^nf(n^s(\lam_j-z_0)),$$ where the $\lam_j$ denote the eigenvalues.
  By the local circular law by Yin \cite[Th. 1.2]{yinLCL3} (see also \cite[Th. 9]{TaoVu} for the case where $|z_0|<1$), we know that with \pro tending to one,
  $$\sum_{j=1}^nf(n^s(\lam_j-z_0))\ge \f n\pi \int_{|z|\le 1}f(n^s(\lam_j-z_0))\ud L(z)-n^{\si},$$ where $L$ denotes the Lebesgue measure on $\C$ and $\si:=(1-2s)/2$.
  We deduce that with \pro tending to one, $$N_s\ge\sum_{j=1}^nf(n^s(\lam_j-z_0))\ge \f {n^{1-2s}}4-n^\si.$$
  \epr

     \blem\la{lem:spacing_eig}  Let $z,z'$ in the unit disk, possibly depending on $n$, \st for a certain fixed $\eps\in (0, 1/2)$, uniformly in $n$, $$\sqrt n |z'-z|\ge n^\eps. $$
Let $\lam, \lam'$ be the eigenvalues of $X$ that are the    closest to respectively  $z$ and $z'$ (if  $\lam$ is the closest eigenvalue to both $z$ and $z'$, then $\lam'$ is the second closest to $z'$). Then for any fixed $\del\in (0, \eps)$, we have  $$\sqrt n |\lam'-\lam|\ge n^\del$$ with \pro tending to one  as $n\to\infty$. \elem

\bpr Let  $s\in (1/2-\eps,1/2)$. By the previous lemma, we have $$|\lam-z|\le n^{-s}\AND |\lam'-z'|\le n^{-s}$$ with \pro tending to one as $n\to\infty$.
Thus $$|\lam'-\lam|\ge |z'-z|-|\lam-z|-|\lam'-z'|\ge n^{-1/2+\eps}-2n^{-s},$$ which allows to conclude, as $-s<-1/2+\eps$ and $ \eps>\del$.
\epr

 \brem In Lemmas \re{lem:spacing_eigprelim} and \re{lem:spacing_eig}, the properties do not only hold with \pro tending to one but  with \pro at least $1-Cn^{-D}$
 for any $D>0$ (and for $C$ a constant depending only on $D$, not on $z_0, z,z'$), which can be useful when using a union bound. The proof is the same and follows from the fact that in \cite{yinLCL3}, the error \pro is $\le Cn^{-D}$. \erem

  \bpr[Proof of \theo{thCVeigvectorsGinibre}]
  For $u,v$ some random variables implicitly depending on $n$, we use the notation $u\sim v$ (resp. $u=O(v)$, $u=o(v)$) when $u/v$ tends in \pro to one (resp.  $u/v$ is tight,  $u/v$ tends in \pro to $0$) as $n\to\infty$.

 Set $\cO:=\cO_{z_1,\ld,z_k,z_k, \ld, z_k}$ (see \eqre{defO} for the definition of this   set).  By Theorem \re{thEigenvectors}, we know that $X$ can be written $X=UTU^*$ with $U$ unitary and $T=[t_{ij}]$ upper triangular having the density
 \be\la{densityTproof} \propto\one_{(t_{11},\ld, t_{nn})\in \cO }|\Del(t_{11},\ld,t_{nn})|^2 \me^{-n\sum_{1\le i\le j\le n}|t_{ij}|^2},\ee so that the random variables \be\la{def:ttilde}\wtt_{ij}:=\sqrt nt_{ij}\ee are  independent  standard complex Gaussian variables.

By definition of
$\cO$, $\lam_1,\ld,\lam_k$ are the $k$ first diagonal entries of $T$.

Besides,  as $U$ is unitary,
\be\la{eq:veqw}\lf(\sqrt n(\lam_j-\lam_i) \lan\bv_i,\bv_j\ran\ri)_{1\le i<j\le k}=\lf(\sqrt n(\lam_j-\lam_i) \lan\bw_i,\bw_j\ran\ri)_{1\le i<j\le k}\ee where the $\bw_i$ are the eigenvectors of $T$ associated to the $\lam_i$ (multiplied by independent uniform phases $\me^{\ii\tta_i}$, independent of $T$).

For each $i$, $\bw_i$ is in the kernel of $T-\lam_i$, hence has only its $i$ first coordinates non zero, and these coordinates are proportional to the vector $(x_i(1), \ld, x_i(i))\in \C^i$, satisfying \beq
(\lam_{1}-\lam_i)x_i(1)+t_{1,2}x_i(2)+\cdots\cdots\cdots\cdots\cdots\cdots\cdots+t_{1,i}x_i(i)&=&0\\
\ddots\qquad \qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad \qquad\vdots\;\;&=&\,\vdots\\
(\lam_{i-2}-\lam_i)x_i(i-2)+t_{i-2,i-1}x_i(i-1)+t_{i-2,i}x_i(i)&=&0\\
(\lam_{i-1}-\lam_i)x_i(i-1)+t_{i-1,i}x_i(i)&=&0\\
x_i(i)&=& 1\eeq
We solve this linear system:
\beq &&x_i(i)\;=\; 1
\\
&&x_i(i-1)\;=\;\ff{\lam_i-\lam_{i-1}}t_{i-1,i}\\
&&x_i(i-2)\;=\;\ff{\lam_i-\lam_{i-2}}\lf(t_{i-2,i-1}x_i(i-1)+t_{i-2,i}\ri)\\
&&x_i(i-3)\;=\;\ff{\lam_i-\lam_{i-3}}\lf(t_{i-3,i-2}x_i(i-2)+t_{i-3,i-1}x_i(i-1)+t_{i-3,i}\ri)\\
&&\;\vdots\\ &&\;\vdots\\
&&x_i(1)\;=\;\ff{\lam_i-\lam_{1}}\lf(t_{1,2}x_i(2)+\cdots+t_{1,i-1}x_{i}(i-1)+t_{1,i}\ri)
\eeq
For each $1\le i<j\le k$, we have \be\la{eq:wxi}\lan \bw_i,\bw_j\ran=\f{\me^{\ii(\tta_j-\tta_i)}}{\|x_i\|\|x_j\|}\sum_{\ell=1}^i\ovl{x_i(\ell)}x_j(\ell).
\ee
To analyse the asymptotic behavior of these inner products, let us analyse the asymptotic behavior of each variable $x_i(\ell)$, $1\le \ell\le i\le k$.

By \eqre{300518111} and  \lemm{lem:spacing_eig}, using the fact that $k$ is fixed, we have
\be\la{eq:maxinv1805}\max_{1\le i<j\le k}\ff{|\lam_j-\lam_i|}=o(n^{\ff 2-\f\eps2}).\ee

By the previous equations and the estimate \eqre{eq:maxinv1805}, using the fact that the random variables  $\wtt_{ij}=\sqrt nt_{ij} $  are independent standard complex Gaussian variables, we have, for any $i=1, \ld, k$, we obtain successively the following estimates:
\beqy\nonumber &&x_i(i)\;=\; 1
\\
\la{300518a}&&x_i(i-1)\;\sim\;\ff{\lam_i-\lam_{i-1}}\f{\wtt_{i-1,i}}{\sqrt n}\;=\;o(n^{-\eps/2})\\
\la{300518b}&&x_i(i-2)\;\sim\;\ff{\lam_i-\lam_{i-2}}\f{\wtt_{i-2,i}}{\sqrt n}\;=\;o(n^{-\eps/2})\\
\nonumber&&\;\vdots\\ \nonumber&&\;\vdots\\
\la{300518c}&&x_i(1)\;\sim\;\ff{\lam_i-\lam_{1}}\f{\wtt_{1,i}}{\sqrt n}\;=\;o(n^{-\eps/2})
\eeqy
It implies that for all $1\le\ell\le i-1$, $$x_i(\ell)=O\lf(\ff{\sqrt n(\lam_i-\lam_\ell)}\ri)$$ and that, as $x_i(i)=1$,
 $$\|x_i\|\sim 1.$$ By \eqre{eq:wxi}, we deduce that
 $$\sqrt n(\lam_j-\lam_i)\lan \bw_i,\bw_j\ran\sim {\me^{\ii(\tta_j-\tta_i)}} \wtt_{ij}+ \ff{\sqrt n}O\lf(\sum_{\ell=1}^{i-1} \f{\lam_j-\lam_i}{\ovl{\lam_i-\lam_\ell}(\lam_j-\lam_\ell)}\ri).$$
 For each $\ell=1, \ld, i-1$,
 $$\f{\lam_j-\lam_i}{\ovl{\lam_i-\lam_\ell}(\lam_j-\lam_\ell)}=\ff{\ovl{\lam_i-\lam_\ell}}-\f{\lam_i-\lam_\ell}{\ovl{\lam_i-\lam_\ell}}\ff{\lam_j-\lam_\ell}=o(n^{\ff 2-\f\eps2}),$$ where we used \eqre{eq:maxinv1805}.
 It follows that $$\sqrt n(\lam_j-\lam_i)\lan \bw_i,\bw_j\ran\sim {\me^{\ii(\tta_j-\tta_i)}} \wtt_{ij}+ o(n^{-\eps/2}),$$
  and, as $(\wtt_{ij})_{1\le i<j\le k}$ is a collection of independent standard complex Gaussian variables independent of the $\tta_i$'s,  the result is proved.
  \epr
 \section{Appendix}
 \subsection{Change of variables in the Schur decomposition}
  We endow the sets $\cM_n(\C)$ and $\cT_n(\C)$ of $n\ti n$   respectively complex matrices and upper-triangular  complex matrices with the Euclidian structures defined by \be\la{def:EuclidianStructure}X\cdot Y:=\Re\Tr(XY^*)\ee and let $\ud M$ (resp. $\ud T$) denote the associated Lebesgue measure on $\cM_n(\C)$ (resp. on $\cT_n(\C)$).   We also denote by $\cU_n$ the group  of unitary $n\ti n$ matrices and by $\ud U$ the Haar measure on $\cU_n$.

 \brem\la{whyeigenvalyesaredistinct} It is useful to note that the set of matrices  in $ \cM_n(\C)$ with multiple eigenvalues is the set of matrices whose characteristic polynomial has null discriminant (see \cite[Def. A.10]{Alice-Greg-Ofer}), so that this set is a level set of a non constant polynomial function on  $ \cM_n(\C)$, hence has zero Lebesgue measure (the last fact can be checked by applying Fubini's theorem).
 \erem

We begin by
defining \emph{admissible  sets}, a notion which will allow us to order the eigenvalues of non Hermitian matrices in quite general ways. $S_n$ denotes the set of permutations of $\ensn$.

  \beg{Def}\la{defadmiset} An open set $\cO\subset\C^n$ is said to be \emph{admissible} if the sets $\si\cdot \cO:=\{(t_{\si(1)}, \ld, t_{\si(n)})\ste (t_1, \ld, t_n)\in \cO\}$, $\si\in S_n$, are pairwise disjoint and $\C^n\bck\cup_{\si\in S_n}\si\cdot \cO$ has null Lebesgue measure.\en{Def}

  An important example of
admissible set is the following one.
Fix $z_1, \ld, z_n\in \C$. Then the set   of $n$-tuples $(t_1, \ld, t_n)\in\C^n$ where for each $i$, the $i$-th entry $t_i$ is  strictly  closer to $z_i$ than all  the forthcoming ones, \ie the set
\be\la{defO} \cO_{z_1, \ld, z_n}:=\{(t_1, \ld, t_n)\in\C^n\ste \forall   i<j, |z_i-t_i|<|z_i-t_j|\},\ee
   is admissible.

 \beg{Th}\la{thEigenvectors}Let $\rho$ be a non negative measurable function on $\cM_n(\C)$ \st for any $M\in \cM_n(\C)$ and any unitary matrix $U$, \be\la{rhoinv}\rho(M)=\rho(UMU^*).\ee Fix an admissible set $\cO\subset \C^n$. Then the measure $\rho(M)\ud M$ on $\cM_n(\C)$ is the push-forward, by the function $(U,T)\mapsto UTU^*$, of the measure $$C_n\ud U\otimes \lf(\one_{(t_{11},\ld, t_{nn})\in \cO}|\Del(t_{11},\ld,t_{nn})|^2\rho(T)\ud T\ri)$$ on $\cU_n\times \cT_n(\C)$, where $$\Del(t_{11},\ld,t_{nn}):=\prod_{1\le i<j\le n}(t_{jj}-t_{ii})$$ and $C_n$ is a constant depending only on $n$ (and not on $\rho$).
 \en{Th}

 \brem Using the case of Ginibre matrices, one can compte $C_n$ : \be\la{Eq:Cn}C_n=\ff{\pi^{\f{3n^2-n}2}\prod_{1\le k\le n-1}k!}
\ee
 \erem

 \bpr[Proof of \theo{thEigenvectors}]   Some statements which are very close to \theo{thEigenvectors} are proved in various texts, as \cite{Ginibre,Mehta,Forrester,TAO2}.  However, firstly, these results are a bit less general and written in slightly different languages and, secondly and more importantly, they do not treat the question of the ordering the diagonal entries of $T$ (which is the cornerstone of our approach in this paper). For this reason, we provide a complete proof.

 \beg{lem}\la{lem_fondamental}Let $\cT, \cU, \cM$ be some open subsets of respectively $\R^p$, $\R^q$ and $\R^{p+q}$. Let $\vfi : \cT\ti \cU\to \cM$ be a smooth diffeomorphism with reciprocal denoted by $\Psi=(\Psi_1, \Psi_2)$. Let also $\rho$ be a non negative measurable function on $\cM$. Let $\ud t, \ud u, \ud m$ denote the Lebesgue measures on respectively $\cT, \cU, \cM$. Then the push-forward of the measure $\rho(m)\ud m$ on $\cM$ by the function $\Psi_1 : \cM\to \cT$ is $K(t)\ud t$, with $$K(t):=\int_{u\in \cU}\rho\circ \vfi(t,u)|J\vfi(t,u)|\ud u,$$ with $|J\vfi(t,u)|$ the Jacobian\footnote{What we call here the \emph{Jacobian} of a smooth  function between two Euclidian spaces with the same dimension is the absolute value of the determinant of the matrix of its derivative in any pair of orthogonal bases.} of $\vfi$.
 \en{lem}

 \bpr Let $f: \cT\to\R$ be a test function. Then \beq \int_{m\in \cM} f\circ \Psi_1(m)\rho(m)\ud m&=&\int_{(t, u)\in \cT\ti\cU} \underbrace{f\circ \Psi_1\circ\vfi(t,u)}_{f(t)}\rho\circ \vfi(t,u)|J\vfi(t,u)|\ud u\ud t\\ &=&\int_{t\in \cT} f(t)K(t)\ud t.\eeq
 \epr

  \beg{Def}\la{defmatsets}  Let $\cU'$ be the set of   unitary matrices whose entries are all non zero, whose diagonal entries are positive and whose principal minors are all invertible. \en{Def}

 The following lemma can be found in \cite[Lemma 2.5.6]{Alice-Greg-Ofer}.

\beg{lem}\la{defTheta}
The map $$\Xi:\cU'\to \R^{  n(n-1)}$$ which maps $U$ to \be\la{1521216h39}\bbm 0& \f{u_{12}}{u_{11}} & \f{u_{13}}{u_{11}}&& \cd&\f{u_{1n}}{u_{11}}\\ \\
&0& \f{u_{23}}{u_{22}}&& \cd&\f{u_{2n}}{u_{22}}\\ \\
&&0&&&\\
&&&\ddots&&\\
&&&&0&\f{u_{n-1,n}}{u_{n-1,n-1}}\\
0&0&0&\cd&0&0
\ebm\ee
(without the zeros) is diffeomorphism from $\cU'$ onto a subset of
 $ \R^{ n(n-1)/2}$ with closed null mass complementary. We denote its inverse by $\Theta$.\en{lem}

 Set  \be\la{def:tcO}\cT_{\cO}:=\{T=[t_{ij}]\in \cT_n(\C)\ste  (t_{11},\ld, t_{nn})\in \cO\}.\ee

\beg{lem}[Schur decomposition]\la{SchurDecLemma}For $ \cU'$ as
in Definition \re{defmatsets}  and $\cT_{\cO}$  as in \eqre{def:tcO},
there is an open subset $\cM'\subset\cM_n(\C)$ with   null mass complementary \st any $M\in \cM'$  can be written in a unique way \be\la{SchurEq}M=UTU^*\ee with $T\in \cT_{\cO}$   and $U\in \cU'$.
\en{lem}
Note that if $M=UTU^*$, then

  \be\la{eq29junerho}\rho(M)=\rho(T).\ee

 \beg{Def}\la{defmatsets3} For $\Tta$ as defined in Lemma \re{defTheta}, let
$\vfi : \cT_{\cO}\ti \R^{n(n-1)}\to \cM'$
be the diffeomorphism defined by $\vfi(T,x)=\Tta(x)T\Tta(x)^{*}$ and let $\Psi=(\Psi_1, \Psi_2)$ be its inverse.\en{Def}
By the unitary invariance of  \eqre{rhoinv}, the proof of the theorem reduces to the proof of the fact that
 the push-forward, by $\Psi_1$, of the measure  $$\rho(M)\ud M$$ is$$
 C_n \one_{(t_{11},\ld, t_{nn})\in \cO}\Del(t_{11},\ld,t_{nn})^2\rho(T)\ud T$$ for $C_n$   a constant depending only on $n$.
 By \eqre{eq29junerho} and Lemma \re{lem_fondamental}, this push-forward is the measure   $$\lf(\one_{(t_{11},\ld, t_{nn})\in \cO}\rho(T)\int_{x\in \R^{n(n-1)}}|J\vfi(T,x)|\ud x\ri)\ud T.$$

Then, the following lemma concludes the proof of  \theo{thEigenvectors}.
\beg{lem}\la{lem:20015181}On $\cT_{\cO}\ti \R^{n(n-1)}$, we have $$|J\vfi(T, x)|= g(x)\prod_{1\le i<j\le n}|t_{jj}-t_{ii}|^2,$$ with  $g(x)$ a measurable function of $x$.\en{lem}

\bpr
Let    $\wTta: \cT_n(\C)\ti \R^{ n(n-1)}\to \cT_n(\C)\ti\cU'$ be defined by  $\wTta(T, x):=(T, \Tta(x))$ and $F: \cT_n(\C)\ti \cU'\to \cM_n(\C)$ be defined by $F(T, U):=UTU^*$ (note that $F$ is defined on a manifold and not on an open subset of an Euclidian space). We have $\vfi=F\circ \wTta$  on $\cT_{\cO}\ti \R^{  n(n-1)}$, so we have $$|J\vfi(T, x)|=|JF(T, \Tta(x))|\ti |J\Tta(x)|,$$ hence it suffices to prove that on $\cT_{\cO}\ti \cU'$, $$|JF(T ,U)|=f(U)\prod_{1\le i<j\le n}|t_{jj}-t_{ii}|^2,$$ where $f(\cdote)$ is a function of $U$.

Note that the tangent space of $\cU'$ at $U$ is  the space $$\op{Tangent}_U(\cU')=\{UH\ste H\in\ii\cH_0\}$$
for
\be\la{defiH0}\ii\cH_0:=\{H=[h_{ij}]\in \cM_n(\C)\ste H^*=-H\trm{ and }\forall i,   h_{ii}=0\}.\ee Note also   that for all $T,R\in \cT_n(\C)$, $U\in \cU'$ and $H\in \op{Tangent}_U(\cU')$, we have
\beq DF(T, U)(R,H)&=&HTU^*+URU^*+UTH^*\\
&=&U(U^*HT+R+TH^*U)U^*\\
&=&U(U^*HT-TU^*H+R)U^*
\eeq
 As the transformation of $\cM_n(\C)$ defined by  $K\mapsto UKU^*$ is orthogonal for any unitary $U$, $|JF(T,U)|$ is the absolute value of the determinant of the matrix, in an orthonormal basis, of the map $ \cT_n(\C) \ti \ii\cH_0\to \cM_n(\C)$ defined by $(R,K)\mapsto KT-TK+R$.
Using the fact that $\cM_n(\C)$ is the orthogonal sum of $\cT_n(\C)$ and of the space \be\la{defiTsl}\cT^{sl}:=\{M=[m_{ij}]\in \cM_n(\C)\ste \forall i\le j, m_{ij}=0\}\ee of strictly lower triangular $n\ti n$ matrices, it is easy to see that the determinant of this map is the one of the   map $\mathfrak{C}_T$ of Lemma \re{2112954} below, which  concludes the proof of \lemm{lem:20015181}.
\epr

\beg{lem}\la{2112954}Let  $\cT^{sl}$ be as in \eqre{defiTsl}, $\ii\cH_0$ be as in \eqre{defiH0} and $\pi:\cM_n(\C)\to \cT^{sl}$ be the canonical  projection. Then for any  $T=[t_{ij}]\in \cT_n(\C)$,  the map $\mathfrak{C}_T:  \ii\cH_0\to \cT^{sl}$ defined by $\mathfrak{C}_T(M):=\pi(MT-TM)$,  has Jacobian $$\prod_{1\le i<j\le n}|t_{jj}-t_{ii}|^2,$$ all spaces being endowed with the Euclidian structure induced by \eqre{def:EuclidianStructure}.
 \en{lem}

 \bpr To prove this lemma, we shall first fix some orthonormal bases of $\ii\cH_0$ and $\cT^{sl}$, order them and then prove that the matrix of $\mathfrak{C}_T$ on these (conveniently ordered) bases is lower triangular  by $2\ti 2$ blocs with diagonal blocs having determinants $|t_{jj}-t_{ii}|^2$, $1\le j<i\le n$.

 Let  us denote the elementary matrices by $E_{ij}$ and let $\cB_{\ii\cH_0}$ be the family $$\ff{\sqrt{2}}(E_{ij}-E_{ji},\ii E_{ij}+\ii E_{ji})_{1\le j<i\le n}$$ and let $\cB_{\cT^{sl}}$ be the family $$(E_{ij},\ii E_{ij})_{1\le j<i\le n}.$$ These are of course orthonormal bases of $\ii\cH_0$ and $\cT^{sl}$. Order the set   $\ensn^2$ with the lexicographical order made out of the reverse natural order on the first component  and the natural order on the second one (for this order, for example, $(1,3)\le (1,4)$ and  $(2,1)\le (1,5)$) and use the induced order on
 the bases $\cB_{\ii\cH_0}$ and $\cB_{\cT^{sl}}$.

 Let $M=[m_{ij}]\in {\ii\cH_0}$  and let $[v_{ij}]:=MT-TM$.
 We have \be\la{fortrisuplexic}v_{ij}=\sum_{k=1}^jm_{ik}t_{kj}-\sum_{k=i}^n t_{ik}m_{kj}=m_{ij}(t_{jj}-t_{ii})+\sum_{k=1}^{j-1}m_{ik}t_{kj}-\sum_{k=i+1}^n t_{ik}m_{kj}\ee
It follows that the matrix of 	$\mathfrak{C}_T$ on  the bases $\cB_{\ii\cH_0}$ and $\cB_{\cT^{sl}}$ (ordered as above) is  lower diagonal by $2\ti 2$ blocs, with $2\ti 2$ diagonal blocs the matrices of the linear maps $\C\to\C$ ($\C $ considered as a real vector space) $m\longmapsto (t_{jj}-t_{ii})\ti m$, $1\le j<i\le n$. The determinant of such a map is $|t_{jj}-t_{ii}|^2$, so the result follows.
 \epr \epr

\subsection{Klein's lemma and consequences}
The following lemma can be found in \cite[Lemma 4.4.12]{Alice-Greg-Ofer}.
\blem[Klein's lemma] For any $f:\R\to \R$ convex and $n\ge 1$, the function $M\mapsto \Tr f(M)$ is convex on the space of $n\ti n$ Hermitian matrices.\elem

 For the previous lemma, we shall deduce the following one.
 \blem\la{lem:2505181} Let $f:\R^+\to\R$ be \st $g(x):=f(x^2)$ is convex  and $n\ge 1$. Then the function $X\mapsto \Tr f(X^*X)$ is convex on the space $\cM_n(\C)$ of $n\ti n$ complex matrices.
\elem
\brem\la{rem:2505181} As a direct consequence, for any fixed $(t_{11}, \ld, t_{nn})\in \C^n$, the function $$(t_{ij})_{1\le i<j\le n}\in \C^{n(n-1)/2}\mapsto \Tr f(T^*T), $$ for $T$ the upper-triangular matrix with   entries $(t_{ij})_{1\le i\le j\le n}$,  is convex.
\erem

\brem\la{rem:2505182} Suppose now, with the notation of the lemma, that for some $\al>0$,    $g(x)-\f\al2{x^2}$ is convex. Then, by the lemma, the  function $X\mapsto \Tr f(X^*X)-\f{\al}2\Tr X^*X$ is convex on the space $\cM_n(\C)$ of $n\ti n$ complex matrices. In the framework of Remark \re{rem:2505181}, as adding a constant to a convex function doesn't break convexity, it implies that the function $$(t_{ij})_{1\le i<j\le n}\in \C^{n(n-1)/2}\mapsto \Tr f(T^*T)-\f{\al}2\sum_{1\le i<j\le n}|t_{ij}|^2$$ is convex.
\erem

\bpr[Proof of \lemm{lem:2505181}] By Klein's lemma, the function $$X\in\cM_n(\C)\mapsto \Tr g \lf(\bpm 0&X\\ X^*&0\epm\ri)$$ is convex. Then, conclude noting that for $M:=\bpm 0&X\\ X^*&0\epm$,  $$\Tr f(X^*X)=\ff2\lf(\Tr f(X^*X)+\Tr f(XX^*)\ri)=\ff2\Tr f(M^2)=\ff2\Tr g(M).$$
\epr

\subsection{Logarithmic Sobolev Inequalities and concentration}
The following lemma, due to Bobkov, Ledoux and Herbst,  gives a sufficient condition for a \pro measure to satisfy a logarithmic Sobolev Inequality (\emph{LSI})  and states one of  its main consequences (see \cite[Sec. 2.3.2]{Alice-Greg-Ofer} for a definition of LSI and   a reference for the following lemma).

\begin{lem}\la{lem:2505182}For any $V: \R^n \to \R\cup\{+\infty\}$  and $\al>0$ \st  $V(x)-\f\al2{\|x\|_2^2}$ is convex,  the \pro measure $P_{V,\R^n}\propto \me^{-V(x)}\ud x$ satisfies a LSI with constant $\al^{-1}$. This implies that
 for any $1$-Lipschitz function $f:\R^n\to \R$ and any $\del>0$, we have
$$P_{V,\R^n}\lf(|f(x)-\E_{P_{V,\R^n}}f|\ge \del\ri)\;\le\;2\me^{-\al\del^2/2}.$$\end{lem}

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