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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 115, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/115\hfil Hopf maximum principle revisited]
{Hopf maximum principle revisited}

\author[J. C. Sabina de Lis \hfil EJDE-2015/115\hfilneg]
{Jos\'e C. Sabina de Lis}

\address{Jos\'e C. Sabina de Lis\newline
 Departamento de An\'{a}lisis Matem\'atico and IUEA,
 Universidad de La Laguna,
 C. Astrof\'isico Francisco S\'anchez s/n, 38203,
 La Laguna, Spain}
\email{josabina@ull.es}

\thanks{Submitted December 29, 2014. Published April l28, 2015.}
\subjclass[2010]{35B50}
\keywords{Maximum principle}

\begin{abstract}
 A weak version of Hopf maximum principle for elliptic equations in divergence
 form  $\sum_{i,j=1}^N\partial_i(a_{ij}(x)\partial_ju)=0$ with H\"older continuous
 coefficients $a_{ij}$ was shown in \cite{FG}, in the two-dimensional case.
 It was also pointed out that this result could be extended to any dimension.
 The objective of the present note is to provide a complete  proof of this
 fact, and to cover operators more general than the one studied in \cite{FG}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

It is well-known that the Hopf maximum principle (see \cite[Lemma 3.4]{GT},
\cite[Theorem II.7]{PW} or \cite[Theorem 2.8.4]{PS} for a
classical statement) does not hold
for linear elliptic equations in divergence form. More precisely,
a function $u\in C^1(\overline{\Omega})$, with $\Omega\subset \mathbb{R}^N$ a smooth domain, is assumed
 to solve in weak sense the elliptic equation
\begin{equation}\label{e:1}
\sum_{i,j=1}^N \frac{\partial}{\partial x_i}\Big(a_{ij}(x)\frac{\partial u}{\partial x_j}\Big)=0
\end{equation}
in $\Omega$, while $u(x)>u(x_0)$ in an inner tangent ball $B\subset \Omega$,
 $x_0\in \partial\Omega\cap {\overline{B}}$ being the
tangency point. Then, a maximum Hopf principle (a ``boundary point lemma'')
holds at $x_0$ if the strict inequality
\begin{equation}
\label{e:2}
\frac{\partial u}{\partial n} < 0
\end{equation}
is satisfied at $x_0$, $n$ standing for the outward unit normal at that point.

A counterexample to this assertion, even when coefficients $a_{ij}$ in \eqref{e:1}
are continuous in $\overline{\Omega}$ was
given in \cite[ Problem 3.9]{GT}   (see also \cite[Section 2.7]{PS};
and a further example in \cite{Naz} for
the case in which coefficients in \eqref{e:1} satisfy $a_{ij}\in L^\infty(\Omega)$).
Moreover, as pointed out in \cite{Naz}, a simpler example than
the one in \cite{GT} can be obtained as follows.
Function $u = \Re \frac{z}{\ln z}$, $z = x+iy$, is harmonic and
negative in the plane domain $\Omega$ enclosed by the $C^1$ curve
$r = \varphi(\theta)$ with $\varphi(\theta) = \text{exp}\ (-\theta \tan \theta)$
if $|\theta|<\frac\pi2$, $\varphi(\pm\pi/2) =0$ (\cite{GT}, Chapter 3).
Outward unit normal at $(0,0)\in \partial\Omega$ is $n = (-1,0)$ while
$$
u(0,0)=0,\quad u_x(0,0)=0.
$$
Thus \eqref{e:2} fails. Since $\Omega$ is not a $C^2$ domain at $(0,0)$ then,
after a $C^1$ rectification of $\partial \Omega$ near
$(0,0)$ one finds that $u$ solves an equation \eqref{e:1} with respect to
new variables $(x',y')$ in $B\cap\{y'>0\}$,
with coefficients $a_{ij}\in C({\overline{B}}\cap\{y'\ge 0\})$, being $B$ a
 small ball centered at $(0,0)$.
This furnishes us the desired counterexample.

Nevertheless, Hopf maximum principle, when regarded in this weak form,
seems to be either not correctly stated
(see e. g. \cite[Proposition 1.16]{DF} where some kind of differentiability
assumption on the coefficients seems to be missing)
or not properly employed in comparison arguments
(proof of \cite[Proposition 2.2]{GV}, Remark \ref{veron} bellow).

The difficulty in showing a Hopf maximum principle for \eqref{e:2} lies,
of course, on the lack of differentiability
of coefficients $a_{ij}$. Indeed, a proof in the line of the standard one
 works provided that the $a_{ij}$ belong
to $C^{0,1}(\overline{\Omega})$. That is why it still seems an outstanding result the
fact that Hopf principle holds when the $a_{ij}$ are merely H\"{o}lder continuous.
This was shown in \cite[Lemma 7]{FG} for \eqref{e:1} in the case
$N=2$ (a later improved two-dimensional statement appeared in \cite{G}).
Moreover, it is asserted in \cite[Remark 2 p. 35]{FG} that:
 ``The proof of Lemma 7 can be extended to $n$ dimensions for equations of the
form \eqref{e:1}''.
Accordingly, the goal of this note is to furnish to the interested reader
a detailed proof of such extended version.
In addition, the operators we are addressing in the present article
 are slightly more general than the one announced in \cite{FG},
meanwhile some of the auxiliary results obtained here seem interesting
in its own right
(see estimate \eqref{ar:7} in Lemma \ref{arl:3} bellow).

To simplify  notation we are using, whenever possible,
either $\partial_i$ or $\partial_{ij}$ instead of  $\frac \partial{\partial x_i}$ or
$\frac {\partial^2}{\partial x_i\partial x_j}$, respectively,
wherein reference variable $x$ could be replaced by another one,
say $y$ depending on the context.

Our main result is stated as follows.

\begin{theorem}\label{thm1}
Let $\Omega\subset \mathbb{R}^N$ be a smooth  bounded domain,
$a_{ij}\in C^\alpha(\overline{\Omega})$ with
$a_{ij}(x)=a_{ji}(x)$, $1\le i,j\le N$, $x\in\overline{\Omega}$,  and 
\begin{equation}\label{i:0}
\sum_{i,j=1}^N a_{ij}(x)\xi_i\xi_j >0,
\end{equation}
for all $x\in\overline{\Omega}$ and $\xi\in\mathbb{R}^N\setminus\{0\}$. Assume that 
$u\in C^1(\overline{\Omega})$
solves, in the weak sense,
\begin{equation}
\label{i:1}
-\sum_{i,j=1}^N \partial_i(a_{ij}(x)\partial_j u) + \sum_{i=1}^N b_i(x)\partial_i u + c(x)u \ge 0,
\end{equation}
in $\Omega$, where $b_i\in L^\infty(\Omega)$ for $1\le i \le N$, 
$c\in L^\infty(\Omega)$ and $c(x)\ge 0$ a. e. in $\Omega$.

Suppose that for $x_0\in \partial\Omega$ there exists a ball $B\subset\Omega$ with 
$x_0\in \partial B$ where $u=u(x)$ satisfies:
$$
u(x) > u(x_0)\quad x\in B.
$$
If $u(x_0)\le 0$ then
\begin{equation}
\label{i:2}
\frac{\partial u}{\partial \nu}(x_0)< 0,
\end{equation}
where $\nu$ is {\rm any} outward direction, i. e. any unitary vector 
$\nu \in \mathbb{R}^N$ so that $\langle \nu, n\rangle >0$, $n$ being the
outward unitary normal at $x_0$.
\end{theorem}

\begin{remark} \rm \begin{itemize}
\item[(a)] Ball $B$ in the statement is indeed an ``inner ball'' tangent to
 $\partial\Omega$ at $x_0\in \partial\Omega$.
\item[(b)] No restriction on the sign of $c$ is required in the case where 
$u(x_0)=0$. Alternatively,
the sign of $u(x_0)$ can be arbitrary provided that $c(x)=0$ for all $x\in \Omega$.
\end{itemize}
\end{remark}

\section{Proof of Theorem \ref{thm1}}\label{dos}
Define the operator,
$$
\mathcal{L} u = -\sum_{i,j=1}^N \partial_i(a_{ij}(x)\partial_j u),
$$
which is understood to act in weak sense on functions in $C^1(\overline{\Omega})$. 
Let $u$ be as in the statement of
Theorem \ref{thm1} and set $u_0 = u(x_0)$. Then
$$
\mathcal{L}(u-u_0)+\sum_{i=1}^N b_i\partial_i (u-u_0)+c(u-u_0)\ge -c u_0\ge 0.
$$
By performing a suitable linear transformation of the variable $x$ it can be 
assumed that $a_{ij}(x_0)=\delta_{ij}$ ($\delta_{ij}=1$ if $i=j$, $\delta_{ij}=0$ otherwise).
 After a translation and a rotation it can be also assumed that $x_0 = 0$ 
meanwhile the outward unit normal $n$ at $x=0$ is $-e_N$. It should be remarked 
that after this set of variable changes, outward derivatives of $u$ at
$x_0$ are transformed into outward derivatives of $u$ at $0$, with respect 
to the new variables.

Consider the ``unitary'' annulus $ D= \{x\in \mathbb{R}^N:1/2 < |x| < 1\}$ and for
$\rho >0$ set $D_\rho = \rho D = \{\rho x: x\in D\}$. By a suitable choice of $\rho_0>0$ it follows that
the domain
$$
\Omega_\rho = \rho e_N+D_{\rho}=\{x:\rho/2 <|x-\rho e_N|<\rho\},
$$
lies in $\Omega$ for all $0 < \rho < \rho_0$.

Following \cite{FG} we introduce the auxiliary function $v\in C^{1,\alpha}(\overline{\Omega}_\rho)$
defined as the weak solution to the problem
\begin{equation}\label{d:1}
\begin{gathered}
\mathcal{L} v + \sum_{i=1}^N b_i\partial_i v + cv = 0 \quad x\in \Omega_\rho\\
 v = 1 \quad x\in \partial\Omega_\rho^-\\
 v = 0 \quad x \in \partial \Omega_\rho^+,
\end{gathered}
\end{equation}
where $\partial\Omega_\rho^- =\{x: |x-\rho e_N|=\rho/2\}$ and
$\partial\Omega_\rho^+ =\{x: |x-\rho e_N|=\rho\}$. Existence and
uniqueness of a positive solution to \eqref{d:1} is provided in
Lemma \ref{lem2} below.

It is clear that a small $\varepsilon>0$ can be found so that
$$
u-u_0-\varepsilon v \ge 0,
$$
on $\partial\Omega_\rho$ meanwhile
$$
\mathcal{L}(u-u_0-\varepsilon v)+c(u-u_0-\varepsilon v) \ge 0,
$$
in the weak sense in $\Omega_\rho$. The weak maximum principle \cite{GT} then 
implies that
$u \ge u_0 + \varepsilon v$,
in $\Omega_\rho$. In particular,
\begin{equation}\label{d:2}
\frac{\partial u}{\partial \nu}(0)\le \varepsilon \frac{\partial v}{\partial \nu}(0),
\end{equation}
for any outward direction $\nu$ to $\Omega_\rho$ at $x=0$. 
It follows from Lemma \ref{arl:3} below that
\begin{equation}
\label{d:3}
\frac{\partial v}{\partial \nu}(0)\to -\infty
\end{equation}
as $\rho\to 0+$. An even more precise account on the asymptotic behavior of 
such derivative as $\rho\to 0+$ is given in Lemma \ref{arl:3}.
It is clear that \eqref{d:2} and \eqref{d:3} imply the desired conclusion 
\eqref{i:2}.

\begin{remark}\label{veron} \rm
The following strong comparison principle is stated in 
\cite[Proposition 2.2]{GV}.  Functions
$u,v\in C^1(\overline{\Omega})$, $u=v=0$ on $\partial\Omega$, solve $-\Delta_p u=f$ and 
$-\Delta_p v=g$ in a smooth bounded
domain $\Omega \subset \mathbb{R}^N$. 
It is assumed that $f,g\in L^\infty(\Omega)$, $0 \le f \le g$ while the set
$\{x\in \Omega: f(x)=g(x)\text{ a. e.}\}$ has an empty interior. 
Then $v(x)>u(x)$ for all $x\in\Omega$ together with
\begin{equation}\label{veronineq}
\frac{\partial v}{\partial n}< \frac{\partial u}{\partial n},
\end{equation}
at every point in $\partial\Omega$. 

As for its proof, by the contradiction argument
 employed in \cite{GV} it follows that $v>u$ in $\Omega$. 
This is achieved by using weak comparison \cite{To} and the strong maximum principle
\cite{V}, the latter implying that $\frac{\partial v}{\partial n}<0$ on $\partial\Omega$.
 Authors in \cite{GV} then obtain
\eqref{veronineq} from the strict inequality between $u$ and $v$ in $\Omega$.

However, we think that to attain \eqref{veronineq} a more  work is required and
propose the following argument. Fix $x_0\in\partial\Omega$
and assume that contrary to \eqref{veronineq} the equality
\begin{equation}
\label{verdos}
\frac{\partial v}{\partial n}(x_0)= \frac{\partial u}{\partial n}(x_0)
\end{equation}
holds. Then there exists a small ball $B$, centered at $x_0$,  such  that
$$
\min\{|\nabla u(x)|,|\nabla v(x)|\}\ge k >0
$$
in $U:= B\cap \Omega$. Thus, the difference $w=v-u$ solves in $U$ an elliptic
equation of the form \eqref{e:1} with the uniform elliptic matrix
\begin{equation}\label{matriz}
A(x) = \int_0^1 |\nabla w_t|^{p-2} 
\Big(I+(p-2)\frac{\nabla w_t}{|\nabla w_t|}\otimes \frac{\nabla w_t}{|\nabla w_t|}\Big)\,dt,
\end{equation}
where $w_t = (1-t)u+t v$, $I$ is the identity matrix and for
 $\xi\in \mathbb{R}^N$, $(\xi\otimes\xi)_{ij}=\xi_i\xi_j$,
$1\le i, j\le N$. Since $\frac{\partial v}{\partial n}(x_0)\neq 0$ this implies, by 
reducing $B$ if necessary,
that $0\notin [\nabla u(x),\nabla v(x)]$ for all $x\in \overline{U}$. 
Equivalently, that
$|\nabla w_t(x)|>0$ for all $x\in \overline{U}$, $t\in [0,1]$. 
Taking into account that $u,v\in C^{1,\alpha}(\overline{\Omega})$ for some 
$0 < \alpha < 1$ \cite{Li} then the coefficients $a_{ij}$ of matrix $A$ in
\eqref{matriz} belong to $C^{\alpha}(\overline{U})$. 
In this respect it should be remarked that $\nabla u(x)\neq 0$ and
$\nabla v(x)\neq 0$ in $\overline{U}$ are not enough to ensure us that
 $a_{ij}\in C^{\alpha}(\overline{U})$.
Finally, Theorem \ref{thm1} can now be used to conclude that \eqref{verdos}
is not possible. Hence, \eqref{veronineq} holds at $x_0$.
\end{remark}


\section{Auxiliary results}\label{ar}

\begin{lemma} \label{lem2}
Problem \eqref{d:1} admits a unique positive solution $v\in C^{1,\alpha}(\overline{\Omega}_\rho)$.
\end{lemma}

\begin{proof}
Existence of a unique weak solution $v\in H^1(\Omega_\rho)$ to \eqref{d:1} 
is standard \cite[Theorem 8.3]{GT},
being the uniqueness consequence of the weak maximum principle. 
Just this result implies that $0 \le v \le 1$
a. e. in $\Omega$. Since $v\in L^\infty(\Omega)$, classical results in \cite{LU} 
imply that $v\in C^\beta(\overline{\Omega}_\rho)$ for some $0 < \beta <1$. Furthermore, 
strong maximum principle \cite[Theorem 8.19]{GT} ensures
us that $v(x) > 0$ for all $x\in \Omega_\rho$. Also the  results in 
\cite[Section 8.11]{GT} permit us concluding that
$v\in C^{1,\alpha}(\overline{\Omega}_\rho)$.
\end{proof}

\begin{remark}\rm 
When $b_i\equiv 0$, $1\le i \le N$, in \eqref{d:1} existence of a weak 
solution can be directly obtained by a variational argument.
 In fact, the functional
$$
J(u) = \frac 12\int_{\Omega_\rho}\Big\{\sum_{i,j=1}^N a_{ij}\partial_i u\partial_j u + c u^2\Big\},
$$
is coercive in $\mathcal M = \{u\in H^1(\Omega_\rho):u=\varphi \ \text{on}\ \partial\Omega_\rho\}$, 
where $\varphi$ is the characteristic function of $\partial\Omega_\rho^-$ 
in $\partial\Omega_\rho$. It therefore admits a global minimizer
$u\in H^1(\Omega_\rho)$ in $\mathcal M$. Moreover, such minimizer is unique due to
the convexity of $J$ ($c\ge 0$).
\end{remark}

Consider now the elliptic operator
$$
\overline{\mathcal{L}} u = -\sum_{i,j=1}^N {\bar{a}}_{ij}\partial_{ij} u,
$$
where the coefficients ${\bar{a}}_{ij}$ are constant and 
the matrix ${\bar{A}} = ({\bar{a}}_{ij})$
is symmetric and positive definite with eigenvalues
$$
0 < \bar{\lambda}_1\le\cdots\le \bar{\lambda}_N.
$$
Let $D$ be the unitary annulus introduced above and $D_\rho$ the corresponding annulus
with exterior radius $\rho$. Set $G_\rho(x,y)$ the Green function
associated to $\overline{\mathcal{L}}$, under homogeneous Dirichlet conditions in 
$D_\rho$ (see \cite{DB}).
Namely, the unique function
$G_\rho\in C^{2}(\overline{D}_\rho\times \overline{D}_\rho\setminus \Delta)$,
$\Delta = \{(x,x):x\in \overline{D}_\rho\}$, such that
the classical solution $u\in C^2({D_\rho})\cap C(\overline{D}_\rho)$ to the problem
\begin{equation}\label{ar:1}
\begin{gathered}
\overline{\mathcal{L}} u = f \quad x\in D_\rho\\
u = 0 \quad x\in \partial D_\rho,
\end{gathered}
\end{equation}
with $f\in C({D_\rho})\cap L^1({D_\rho})$, provided that it exists, 
can be represented in the form
\begin{equation}\label{ar:2}
u(x)= \int_{D_\rho}G_\rho(x,y)f(y)\,dy.
\end{equation}

The next result provides us with the key estimates on the derivatives of $G_\rho$.

\begin{lemma}\label{arl:1} 
There exist positive constants $C_1$, $C_2$ such that
\begin{gather}\label{ar:3}
|\partial_{x_i}G_\rho(x,y)|\le \frac{C_1}{|x-y|^{N-1}}\quad 1 \le i \le N, \\
\label{ar:4}
|\partial_{x_i}\partial_{y_j}G_\rho(x,y)|\le \frac{C_2}{|x-y|^{N}}\quad 1 \le i, j \le N,
\end{gather}
for all $x,y\in D_\rho$, $x\neq y$. Moreover, constants $C_1$ and $C_2$ 
can be estimated as follows:
\begin{equation} \label{ar:5}
C_1 \le K_1 \Big( \frac{\bar{\lambda}_N}{\bar{\lambda}_1}\Big)^{\frac{N-1}2}
\frac 1{\bar{\lambda}_1}, \qquad
C_2 \le K_2 \Big( \frac{\bar{\lambda}_N}{\bar{\lambda}_1}\Big)^{\frac{N}2}\frac 1{\bar{\lambda}_1},
\end{equation}
where the positive constants $K_1$ and $K_2$ do not depend on $\rho$.
\end{lemma}

\begin{proof} 
There exists a linear isomorphism $y = Tx$ which maps $D_\rho$ into the
ellipsoidal cavity $\mathcal{D}_\rho = \{\rho y:y\in \mathcal{D}\}$ with
$$
\mathcal{D} = \{y\in \mathbb{R}^N: \frac 14 < \sum_{i=1}^N\frac{y_i^2}{a_i^2}< 1\},
$$
and where the reference semiaxis $a_i$ are given by
$$
a_i = \frac 1{\sqrt{\bar{\lambda}_i}}\quad i = 1,\dots,N.
$$
Moreover, $T$ transforms problem \eqref{ar:1} into
\begin{equation}\label{ar:4b}
\begin{gathered}
-\Delta v = g \quad y \in \mathcal{D}_{\rho}\\
\ v = 0 \quad y\in \partial \mathcal{D}_{\rho},
\end{gathered}
\end{equation}
where $v(y) = u(T^{-1}y)$, $g(y) = f(T^{-1}y)$. 
Let $\widetilde{G}_\rho = \widetilde{G}_\rho(y,\eta)$ be
the Green function associated to $-\Delta$ under homogeneous Dirichlet 
conditions in $\mathcal{D}_\rho$. A direct computation shows that
$$
G_\rho(x,\xi) = \{\det  T\} \ \widetilde{G}_\rho(Tx,T\xi),
$$
for all $x,\xi\in D_\rho$, $x\neq \xi$, where
$\det T = a_1\cdots a_N$.
A further scaling argument permits us writing
$$
\widetilde{G}_\rho(y,\eta) = \rho^{2-N}G\Big(\frac y\rho,\frac \eta\rho\Big),\quad 
y,\eta\in \mathcal{D}_\rho,\; y\neq \eta,
$$
$G = G(z,\zeta)$ being the Green function for $-\Delta$, constrained with 
homogeneous Dirichlet conditions in $\mathcal{D}$.
Therefore,
$$
G_\rho(x,\xi) = \rho^{2-N}\{\det T\}  G\Big(
\frac {Tx}\rho,\frac {T\xi}\rho\Big)\quad x,\xi\in D_\rho\quad x\ne \xi.
$$
Now, the estimates in \cite{Wi} allow us assert the existence of a positive 
constant $M$ such that
\begin{equation}\label{ar:6}
\begin{gathered}
|\partial_{x_i}G(x,y)|\le \frac M{|x-y|^{N-1}}\quad 1 \le i \le N,\\
|\partial_{x_i}\partial_{y_j}G(x,y)|\le \frac M{|x-y|^{N}}\quad 1 \le i,j \le N,
\end{gathered}
\end{equation}
for all $x,y\in \mathcal{D}$, $x\neq y$. Next we observe that isomorphism $T$ 
can be chosen of the form
$$
T = \operatorname{diag}(a_1,\dots,a_N)\ L,
$$
where $L$ is an orthogonal transformation. Thus,
$$
\partial_{x_i}G_\rho(x,y) = \rho^{1-N}\sum_{k=1}^N \partial_{z_k}G
\Big(\frac{Tx}\rho,\frac{Ty}\rho\Big) \partial_{x_i}((Tx)_k),
$$
where
$(Tx)_k = a_k\sum L_{xs}x_s$.
Then, the estimate
$$
\sum_{k=1}^N |\partial_{x_i}((Tx)_k)|\le \sqrt N a_1
$$
follows easily. In addition,
$$
|Tx| = |\operatorname{diag}(a_1,\dots,a_N) Lx|\ge a_1|x|,
$$
for all $x\in \mathbb{R}^N$. By  \eqref{ar:6}  with the last estimates, 
the first inequality in \eqref{ar:5} is obtained with the choice
$$
K_1 = M \sqrt N.
$$
By proceeding in the same way, the second inequality in \eqref{ar:5} holds for
$K_2 = M N$.
\end{proof}

\begin{lemma}\label{arl:3} 
Let $v\in C^{1,\alpha}(\overline{\Omega}_\rho)$ be the positive solution of the problem \eqref{d:1}.
Then,
\begin{equation}
\label{ar:7}
\frac{\partial v}{\partial \nu}(0)\sim \frac{C_N^*}\rho\langle \nu,e_N\rangle \quad \text{as}\quad \rho\to 0+,
\end{equation}
where $\nu\in \mathbb{R}^N$ is any unitary vector and
$$
C_N^*= \frac{N-2}{2^{N-2}-1}.
$$
\end{lemma}

\begin{remark} \rm 
Observe that exterior directions $\nu$ to $\Omega_\rho$ at $x=0$ are
characterized by $\langle\nu,e_N\rangle< 0$.
\end{remark}

\begin{proof}[Proof of Lemma \ref{arl:3}]
To prove \eqref{ar:7} we follow the argument in \cite{FG} and introduce
 $u=\psi$, the solution of the problem
\begin{gather*}
\Delta u = 0\quad x\in \Omega_\rho\\
 u = 1 \quad x\in \partial\Omega_\rho^-\\
 u = 0 \quad x \in \partial \Omega_\rho^+,
\end{gather*}
whose explicit form is 
$$
\psi(x) = \Big( \frac 1{|x-\rho e_N|^{N-2}}-\frac 1{\rho^{N-2}}\Big)
\frac{\rho^{N-2}}{2^{N-1}-1}.
$$
We fix now $\bar x\in \Omega_\rho$ and define the constant coefficients operator
$$
\mathcal{L}_{\bar x} u := -\sum_{i,j=1}^N {\bar{a}}_{ij}\partial_{ij}u,
$$
with ${\bar{a}}_{ij} = a_{ij}(\bar x)$. By noticing that
$w(x):= v(x)-\psi(x)$  vanishes at the boundary $\partial \Omega_\rho$ 
of $\Omega_\rho$, $w$ can be represented as
\begin{equation}\label{d:4}
w(x) = \int_{\Omega_\rho}G_\rho(x,y)\mathcal{L}_{\bar x} w(y)\,dy,
\end{equation}
where $G_\rho$ stands for the Green function of the operator
 $\mathcal{L}_{\bar x}$ in $\Omega_\rho$,
subject to homogeneous Dirichlet conditions on $\partial\Omega_\rho$
 (see Lemma \ref{arl:1}).
We are employing \eqref{d:4} to analyze $\nabla w$ near zero when $\rho$ becomes
small.

Observe that,
\begin{align*}
w(x) &= \int_{\Omega_\rho}G_\rho(x,y)(\mathcal{L}_{\bar x}v(y)-\mathcal{L} v(y))\,dy\\
&\quad - \int_{\Omega_\rho}G_\rho(x,y)(\mathcal{L}_{\bar x}\psi(y)-\mathcal{L}_0\psi(y))\,dy\\
&\quad -\int_{\Omega_\rho}G_\rho(x,y)(b(y)\nabla v(y)+c(y)v(y))\,dy \\
&=: w_1(x)+w_2(x)+w_3(x),
\end{align*}
$x\in\Omega_\rho$, with $b=(b_i)$ and where $\mathcal{L}_0$ is the constant coefficients
 operator resulting from fixing $x=0$
in the functions $a_{ij}(x)$. Notice that $\mathcal{L}_0 = -\Delta$ and 
so $\mathcal{L}_0 \psi = 0$.

On the other hand,
$$
w_1(x) = \sum_{i,j=1}^N\int_{\Omega_\rho}\partial_{y_i}G_\rho(x,y)(a_{ij}(y)
-a_{ij}(\bar x))\partial_jv(y)\,dy.
$$
Hence,
\begin{equation}
\label{ar:8}
\partial_{x_s}w_1(\bar x) = \sum_{i,j=1}^N\int_{\Omega_\rho}\partial_{x_s}\partial_{y_i}G_\rho(\bar x,y)(a_{ij}(y)-a_{ij}(\bar x))\partial_jv(y)\,dy.
\end{equation}
By estimate \eqref{ar:4} in Lemma \ref{arl:1},
$$
|\partial_{x_s}w_1(\bar x)| \le \sum_{i,j=1}^N C_2 [a_{ij}]_\alpha
 \|\nabla v\|_{\infty,\Omega_\rho}\int_{\Omega_\rho}\frac 1{|y-\bar x|^{N-\alpha}}\,dy,
$$
where
$$
[a_{ij}]_\alpha = \sup_{x,y\in\Omega, x\neq y}\frac{|a_{ij}(x)-a_{ij}(y)|}{|x-y|^\alpha}.
$$
After estimating the integral, \eqref{ar:8} implies that
\begin{equation}
\label{ar:9}
|\nabla w_1(\bar x)|\le C \|\nabla v\|_{\infty,\Omega_\rho}\rho^\alpha\quad \bar x\in \Omega_\rho,
\end{equation}
for a certain positive constant $C$ which does not depend on $\rho$.
Label $C$ will be employed in the sequel to designate positive constants
which no depend on $\rho$, and whose precise value is irrelevant for the discourse.

As for the gradient of $w_2$,
$$
\partial_{x_s}w_2(\bar x) = \sum_{i,j=1}^N\int_{\Omega_\rho}\partial_{x_s}
G_\rho(\bar x,y)(a_{ij}(0)-a_{ij}(\bar x))\partial_{ij}\psi(y)\,dy.
$$
Since $|\partial_{ij}\psi(y)|\le C\rho^{-2}$,  using estimate \eqref{ar:3} we find that
$$
|\partial_{x_s}w_2(\bar x)| \le \sum_{i,j=1}^N C [a_{ij}]_\alpha \rho^{\alpha-2}\int_{\Omega_\rho}\frac 1{|y-\bar x|^{N-1}}\,dy.
$$
By estimating the integral in terms of $\rho$ we obtain
\begin{equation}
\label{ar:10}
|\partial_{x_s}w_2(\bar x)| \le C \rho^{\alpha-1}\quad \bar x \in \Omega_\rho.
\end{equation}
On the other hand,  taking into account that $v(0)=0$, we conclude that
$$
|\partial_{x_s}w_3(\bar x)|\le C_1 \|c\|_{\infty,\Omega}\|\nabla v\|_{\infty,\Omega_\rho}
\int_{\Omega_\rho}\frac 1{|y-\bar x|^{N-1}}\,dy,
$$
and so,
\begin{equation}
\label{ar:11}
|\partial_{x_s}w_3(\bar x)| \le C \|\nabla v\|_{\infty,\Omega_\rho}\rho^{2}\quad 
\bar x \in \Omega_\rho.
\end{equation}
From \eqref{ar:9}, \eqref{ar:10} and \eqref{ar:11} the estimate
\begin{equation}
\label{ar:12}
\|\nabla w\|_{\infty,\Omega_\rho}\le C\|\nabla v\|_{\infty,\Omega_\rho}
\rho^{\alpha}+ C\rho^{\alpha-1}
\end{equation}
holds.

Now, $\|\nabla v\|_{\infty,\Omega_\rho}$ can be estimated in terms of $\rho$. In fact,
$$
|\nabla v (x)|\le |\nabla w(x)|+|\nabla \psi(x)|
\le \|\nabla w\|_{\infty,\Omega_\rho}+ C\rho^{-1}\quad x\in \Omega_\rho.
$$
Hence,
$$
\|\nabla v\|_{\infty,\Omega_\rho}\le C \rho^{-1},
$$
which, together with \eqref{ar:12}, imply that
\begin{equation} \label{ar:13}
\|\nabla w\|_{\infty,\Omega_\rho}\le C\rho^{\alpha-1}.
\end{equation}
Finally,
$$
\big|\frac{\partial v}{\partial \nu}(0)-\frac{\partial \psi}{\partial \nu}(0)\big | 
= \big |\frac{\partial v}{\partial \nu}(0)-\frac{C_N^*}\rho\langle\nu,e_N\rangle \big|
\le \|\nabla w\|_{\infty,\Omega_\rho}\le C\rho^{\alpha-1}.
$$
Thus,
$$
\frac{C_N^*}\rho \Big(\langle\nu,e_N\rangle-\frac C{C_N^*}\rho^\alpha\Big) 
\le \partial_{\nu} v(0)
\le \frac{C_N^*}\rho\Big(\langle\nu,e_N\rangle+\frac C{C_N^*}\rho^\alpha\Big),
$$
for $\rho$ small. Asymptotic estimate \eqref{ar:7} immediately 
follows from these inequalities.
\end{proof}

\subsection*{Acknowledgements}
This research was supported by Spanish Ministerio de Ciencia e Innovaci\'on 
and Ministerio de Econom\'{\i}a y
Competitividad under grant reference MTM2011-27998.

\begin{thebibliography}{00}

\bibitem{DF}  D. G. de Figueiredo; 
\emph{Positive solutions of semilinear elliptic problems}.
 Differential equations (S\~{a}o  Paulo, 1981), pp. 34--87,
 Lecture Notes in Math., 957, Springer, Berlin--New York, 1982.

\bibitem{DB}  E. DiBenedetto;
\emph{Partial differential equations}. Birkh\"{a}user Boston, Inc., Boston, MA, 1995.

\bibitem{FG}  R. Finn, D. Gilbarg; 
\emph{Asymptotic behavior and uniqueness of plane subsonic flows}.
Comm. Pure Appl. Maths. \textbf{10} (1957), 23--63.

\bibitem{G}  D. Gilbarg; 
\emph{Some hydrodynamic applications of function theoretic properties of 
elliptic equations}.
Math. Z. 72 1959/1960 165--174.

\bibitem{GT} D. Gilbarg, N. S. Trudinger;
\emph{Elliptic partial differential equations of second order}. Springer-Verlag,
1983.

\bibitem{GV}  M. Guedda, L. V\'{e}ron; 
\emph{Quasilinear elliptic equations involving critical Sobolev exponents}.
Nonlinear Anal. 13 (1989), no. 8, 879--902.

\bibitem{LU}  O. A. Ladyzhenskaya, N. N. Ural'tseva;
\emph{Linear and quasi-linear elliptic equations}. Academic Press,
New York-London, 1968.

\bibitem{Li}  G. M. Lieberman; 
\emph{Boundary regularity for solutions of degenerate elliptic equations.}
Nonlinear Anal. 12 (1988), no. 11, 1203--1219.

\bibitem{Naz}  A. I. Nazarov; 
\emph{A centenial of the Zaremba--Hopf--Oleinik lemma}.
SIAM J. Math. Anal. \textbf{44} (2012), 437--453.

\bibitem{PW}  M. H. Protter, H. F. Weinberger;
\emph{Maximum principles in differential equations}.
Springer-Verlag, New York, 1984.

\bibitem{PS}  P. Pucci, J. Serrin;
\emph{The maximum principle. Progress in Nonlinear Differential Equations
and their Applications}, 73. Birkhäuser Verlag, Basel, 2007.

\bibitem{To}  P. Tolksdorf; 
\emph{On the Dirichlet problem for quasilinear equations in domains with 
conical boundary points}.  Comm. Partial Differential Equations 8 (1983), 
no. 7, 773--817.

\bibitem{V}  J. L. V\'{a}zquez; 
\emph{A strong maximum principle for some quasilinear elliptic equations}. 
Appl. Math. Optim. 12 (1984), no. 3, 191--202.

\bibitem{Wi}  K. O. Widman; 
\emph{Inequalities for the Green functions and boundary continuity of
the gradient of solutions to elliptic equations}.
 Math. Scand. \textbf{21} (1967), 17--37.

\end{thebibliography}

\end{document}