\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
2004-Fez conference on Differential Equations and Mechanics \newline
{\em Electronic Journal of Differential Equations},
Conference 11, 2004, pp. 81--93.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{81}

\begin{document}

\title[\hfilneg EJDE/Conf/11 \hfil An infinite-harmonic analogue]
{An infinite-harmonic analogue of a Lelong theorem
 and infinite-harmonicity cells}

\author[Mohammed Boutaleb\hfil EJDE/Conf/11 \hfilneg]
{Mohammed Boutaleb}

\address{D\'{e}p. de Math\'{e}matiques. Fac de
Sciences F\`{e}s D. M, B.P. 1796 Atlas Maroc}
\email{mboutalebmoh@yahoo.fr}


\date{}
\thanks{Published October 15, 2004.}
\subjclass[2000]{31A30, 31B30, 35J30}
\keywords{Infinite-harmonic functions;  holomorphic extension; 
harmonicity cells;  p-Laplace equation;  stationary plane flow}


\begin{abstract}
 We consider the problem of finding a function $f$ in
 the set of $\infty$-harmonic functions,  satisfying 
 \[
 \lim_{w\to \zeta } |\widetilde{f}(w)| =\infty,\quad
 w\in \mathcal{H}(D),\quad \zeta \in \partial \mathcal{H}(D)
 \]
 and being a solution to the quasi-linear parabolic equation
 \[
 u_x^2u_{xx}+2u_xu_yu_{xy}+u_y^2u_{yy}=0\quad \mbox{in } D\subset
 \mathbb{R}^2\,,
 \]
 where $D$ is a simply connected plane domain,
 $\mathcal{H}(D)\subset \mathbb{C}^2$
 is the harmonicity cell of $D$, and $\widetilde{f}$ is the holomorphic
 extension of $f$.
 As an application, we show a $p$-harmonic behaviour of the modulus of the
 velocity of an arbitrary stationary plane flow near an extreme point of the
 profile.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}

\section{Introduction}

The complexification problems for partial differential equations
in a domain $\Omega \subset \mathbb{R}^n$ include the
introduction of a common domain $\widetilde{\Omega }\supset \Omega $ in
$\mathbb{C}^n$ to which all the solutions of a specified p.d.e. extend
holomorphically. The complex domains in question are the
so-called harmonicity cells  $\mathcal{H}(\Omega )$, in \cite{a4}, for the
following set of $2m$-order elliptic operators:
\begin{equation}
\Delta ^mu=\sum_{|\alpha |=m} \frac{m!}{\alpha !}
\frac{\partial ^{2|\alpha |}u}{\partial x_1^{2\alpha _1}\dots\partial
x_n^{2\alpha _n}}=0,\quad m=1,2,3\dots  \label{e2}
\end{equation}
They often describe properties of physical processes which are
governed by such a p.d.e \cite{o1}. The operator $\Delta ^2$ has been widely
studied in the literature, frequently  in the contexts of biharmonic functions
\cite{a3}.

\subsection*{Motivation}

Our objective is to introduce the complex
domain $\widetilde{D}$, and the adequate solution $f=f_\zeta$ in the
space of $\infty$-harmonic functions ${\bf H}_\infty (D)$ , for
equation \eqref{e6}, below.
In view of Theorem \ref{thm2.5}, part 2, we assign a domain
$\widetilde{D}\subset \mathbb{C}^2$, denoted by $\mathcal{H}_\infty (D)$,
to the class ${\bf H}_\infty (D)$. The definition of $\mathcal{H}_\infty (D)$,
is similar to the definition  of $\mathcal{H}(D)$, although less explicit.
Equation \eqref{e6} is actually the formal limit, as $p\to +\infty $, of the
$p$-harmonic equation in $D\subset \mathbb{R}^2$
\begin{equation}
\Delta _pu=\mathop{\rm div}(|\nabla u|^{p-2}\Delta u)=0,\quad 1<p<+\infty, \label{e3}
\end{equation}
For every finite real $p>1$, the hodograph method transforms
$\Delta_pu=0$ into a linear elliptic p.d.e. in the hodograph plane.
Due to  \cite{b1}, the pull-back operation is possible from $\mathbb{R}^2(u_x,u_y)$
to the physical plane. Although linear, the obtained equation is not easily
computed since its limit conditions become more complicated.

\subsection*{Preliminaries}
Let $\Omega $ be a domain in  $\mathbb{R}^n$, $n\geq 2$,
$\Omega \neq \emptyset$, $\partial \Omega \neq \emptyset $.
In 1935, Aronszajn \cite{a3} introduced the notion of harmonicity cells in
order to study the singularities of $m$-polyharmonic functions. These
functions, used in elasticity calculus of plates, are $C^\infty $-solutions
in $\Omega $ of \eqref{e2}.
Recall that $\mathcal{H}(\Omega )$ is the domain of $\mathbb{C}^n$, whose
trace $\mathop{\rm Tr}\mathcal{H}(\Omega )$ on $\mathbb{R}^n$ is $\Omega $, and
represented by the connected component containing $\Omega $ of the open set
$\mathbb{C}^n-\cup_{t\in \partial \Omega }\Gamma (t)$, where
$\Gamma (t)=\{z\in \mathbb{C}^n:(z_1-t_1)^2+\dots+(z_n-t_n)^2=0\}$ is the
isotropic cone of $\mathbb{C}^n$, with vertex $t\in \mathbb{R}^n$.
Lelong \cite{l1} proved that $\mathcal{H}(\Omega )$ coincides with the set of
 points $z\in \mathbb{C}^n$ such that there exists a path $\gamma $ satisfying:
 $\gamma (0)=z$, $\gamma (1)\in \Omega $ and $T[\gamma (\tau )]\subset \Omega $
for every $\tau $ in $[0,1]$, where $T$ is the Lelong transformation,
mapping points $z=x+iy\in \mathbb{C}^n$ to Euclidean $(n-2)$-spheres
$S^{n-2}(x,\|y\|)$ of the hyperplane of $\mathbb{R}^n$ defined by:
$\langle t-x,y\rangle =0$. If $\Omega $ is starshaped at
$a_0\in \Omega ,\mathcal{H}(\Omega )=\{z\in \mathbb{C}^n;T(z)\subset \Omega \}$
is also starshaped at  $a_0$. Furthermore, for bounded convex domains
$\Omega $ of $\mathbb{R}^n$, we get
\begin{equation}
\mathcal{H}(\Omega )=\big\{z=x+iy\in \mathbb{C}^n:
\max_{t\in T( iy)}{\max }
\big[ \max_{\xi \in S^{n-1}}\big( \langle x+t,\xi \rangle-
\max_{s\in \Omega }\langle \xi ,s\rangle \big) \big] <0\big\}
\end{equation} \label{e4}
where $S^{n-1}$ is the Euclidean unit sphere of $\mathbb{R}^n$
\cite{a4,b2}. The harmonicity cell of the Euclidean unit ball $B_n$ of
$\mathbb{R}^n$
gives a central example, since $\mathcal{H}(B_n)$ coincides with the Lie ball
$LB=\{z\in \mathbb{C}^n;L(z)=[\|z\|^2+\sqrt{\|
z\|^4-|z_1^2+\dots+z_n^2|^2}]^{1/2}<1\}$, where
$\|z\|=(|z_1|^2+\dots+|z_n|^2)^{1/2}$. Besides,
representing also the fourth type of symmetric bounded homogenous
irreducible domains of $\mathbb{C}^n$, $\mathcal{H}(B_n)$ has been studied
(specially in dimension $n=4)$ by theoretical physicits interested in a
variety of different topics: particle physics, quantum field theory, quantum
mechanics, statistical mechanics, geometric quantization, accelerated
observers, general relativity and even harmony and sound analysis (For more
details, see \cite{c1,m1,o1,p1}.

 From the point of view of complex analysis, Jarnicki \cite{j1} proved that if
$D_1$ and $D_2$ are two analytically homeomorphic plane domains of
$\mathbb{C}\simeq \mathbb{R}^2$ then their harmonicity cells
$\mathcal{H}(D_1)$ and $\mathcal{H}(D_2)$ are also analytically homeomorphic
in $\mathbb{C}^2$. A generalization
in $\mathbb{C}^n,n\geq 2$, of this Jarnicki Theorem is established by the
author \cite{b4}, as well as a characterization of polyhedric harmonicity cells
in $\mathbb{C}^2$ \cite{b6}. Furthermore, recall that if ${\bf A}(\Omega )$ and
${\bf Ha}(\Omega )$ denote the spaces of all real analytic and harmonic
functions (respectively) in $\Omega $, then $\mathcal{H}(\Omega )$ is
characterized by the following feature
\begin{equation}
[ \cap_{f\in {\bf Ha}(\Omega )} \Omega ^f]^0=\mathcal{H} (\Omega ), \label{e5}
\end{equation}
while $[\cap \Omega ^f]^0=\emptyset $, when $f$ runs
through ${\bf A}(\Omega )$, where $\Omega ^f$ is the greatest domain of
$\mathbb{C}^n$ to which $f$ extends holomorphically.
We emphasize that in \eqref{e5}, $\Omega $ is actually required to be
star-shaped at some point $a_0$, or a $C$-domain
(that is, $\Omega $ contains the convex hull $\mathop{\rm Ch}(S^{n-2})$ of
any $(n-2)$-Euclidean sphere $S^{n-2}$ included in $\Omega $) or
$\Omega \subset \mathbb{R}^{2p}$ with $2p\geq 4$,
or $\Omega $ is a simply connected domain in $\mathbb{R}^2$ (cf. \cite{a4}).
The technique of holomorphic extension,
used for harmonic functions in \cite{s1}, has been generalized for solutions of
partial differential equations with constant coefficients by Kiselman \cite{k1}.
In a recent paper, Ebenfelt \cite{e1} considers the holomorphic extension to
the so-called kernel $\mathcal{NH}(\Omega )$ of $\Omega $'s
harmonicity cell, for solutions in simply connected domains $\Omega $ in
$\mathbb{R}^n$, of linear elliptic partial differential equations of type:
$\Delta ^ku+\sum_{|\alpha |<2k} a_\alpha (x)D^\alpha u=g$,
where $\mathcal{NH}(\Omega )=\{z\in \mathcal{H}(\Omega );
\mathop{\rm Ch}[T(z)]\subset \Omega \}$. It can be observed that one of
the central results in the theory
of harmonicity cells is the following Lelong theorem (stated here in the
harmonic case)

\begin{theorem} \label{thmA}
 Let $\Omega$ be a non empty domain in
$\mathbb{R}^n$, $n\geq 2$, with non empty boundary and
$\mathcal{H}(\Omega )$ its harmonicity cell in $\mathbb{C}^n$.
For every $\zeta \in\partial \mathcal{H}(\Omega )$ there exists
$f=f_\zeta $, a harmonic function in $\Omega$, which is the restriction to
$\Omega=\mathcal{H}(\Omega )\cap \mathbb{R}^n$ of a (unique)
holomorphic function $\widetilde{f_\zeta }$ defined in
$\mathcal{H}(\Omega )$ such that $\widetilde{f_\zeta }$ can not be extended
holomorphically in any open neighborhood of $\zeta $.
\end{theorem}

\subsection*{Statement of the problem}
In this paper we consider the
simpler case of a non-empty plane domain $D$
(with $\partial D\neq \emptyset$) which we set to be simply connected
and look for a suitable $\infty$-harmonic function $f_\zeta $ in $D$.
We state the problem as follows:

 Let $\zeta $ be a boundary point of $\mathcal{H}(D)$ and put
 $T(\zeta )=\{\zeta_1+i\zeta _2,\bar{\zeta}_1+i\bar{\zeta}_2\}$.
We will assume first that $\zeta $ belongs to $\Gamma (\zeta _1+i\zeta _2)$.
The problem is to find a solution $f_\zeta$ in the classical sense, i.e.
$f_\zeta \in C^2(D)$ and $f_\zeta $ a.e. continuous on $\partial D$ of the
quasi-elliptic system:
\begin{gather}
u_{x_1}^2u_{x_1x_1}+2u_{x_1}u_{x_2}u_{x_1x_2}+u_{x_2}^2u_{x_2x_2}=0\quad
\mbox{in} D \label{e6}\\
\frac \partial {\partial \bar{w}_j}\widetilde{u}=0\quad j=1,2\quad
\quad \mbox{in }\mathcal{H}(D) \label{e6.1}\\
\lim_{w\to \zeta ,\; w\in \mathcal{H}(D)} |\widetilde{u}(w)|=\infty \,.\label{e6.2}
\end{gather}
This problem has already been considered in \cite{l1} in the harmonic case, and
in \cite{b3} in the $p$-polyharmonic case. It has also been solved in the (non
linear) $p$-harmonic case with $1<p<+\infty $ and $n=2$ \cite{b5}. We used in
\cite{b5} radial $p$-harmonic functions and their stream functions, centered at
points of $\partial D$; but this approach limited our results to {\it finite}
real $p$ (with $p>1$) and to {\it real valued} $p$-harmonic functions.
Our main result in the present paper consists of introducing
infinite-harmonicity cells and proving an existence theorem for the
$\infty$-Laplace equation.
In Theorem \ref{thm2.5}, we prove that to $\zeta \in \partial \mathcal{H}(D) $
corresponds a $f_\zeta \in {\bf H}_\infty (D)$ such that
$\widetilde{f_\zeta }$ is holomorphic in $\mathcal{H}(D)$ and satisfies
$|\widetilde{f_\zeta }(w)|\to \infty $, when $w\to \zeta $ with $w$
inside $\mathcal{H}(D)$.


\section{Infinite-harmonicity cells}

The next four propositions are used in this work and their proofs are
found in the references as cited.

\begin{proposition}[\cite{l1}] \label{prop2.1}
Let $\Omega$ be a domain in $\mathbb{R}^n$, $n\geq 2$, $\Omega \neq \emptyset$,
$\partial \Omega \neq \emptyset$, and
$\mathcal{H}(\Omega )\subset \mathbb{C}^n$  be its harmonicity
cell. For every point $\zeta \in \partial \mathcal{H}(\Omega )$,  the
topological boundary of $\mathcal{H}(\Omega )$, one can associate a
point $t\in \partial \Omega$, the topological boundary of $\Omega$,
 such that $\zeta \in \Gamma (t)$,  the isotropic cone of
$\mathbb{C}^n$  with vertex $t$.
\end{proposition}

\begin{proposition}[\cite{a2,l2}] \label{prop2.2}
 A classical solution $u=u(x_1,x_2)\in \mathbf{C}^2$ of the partial
 differential equation
$$
\Delta _\infty u=u_{x_1}^2u_{x_1x_1}+2u_{x_1}u_{x_2}u_{x_1x_2}+u_{x_2}^2u_{x_2x_2}=0,
$$
in every non-empty domain $D\subset \mathbb{R}^2$, is real
analytic in $D$, and  cannot have a stationary point without
being constant
\end{proposition}

\begin{proposition}[\cite{a4}] \label{prop2.3}
To every couple $(\Omega ,f)$, where $\Omega $ is an open set of
$\mathbb{R}^n=\{x+iy\in \mathbb{C}^n;y=0\}$
(equipped with the induced topology from $\mathbb{C}^n$),
$f$ is a real analytic function on $D$, one can
associate a couple $(\widetilde{\Omega },\widetilde{f})$ such that
$\widetilde{\Omega }$ is an open set of $\mathbb{C}^n$ whose trace
$\widetilde{\Omega }\cap \mathbb{R}^n$ with $\mathbb{R}^n$
is the starting domain $\Omega$, and $\widetilde{f}$ is a
holomorphic function in $\widetilde{\Omega }$ whose restriction
$\widetilde{f}|\Omega $ to $\Omega $ coincides with $f$.
 Furthermore, (i) if $\Omega $ is connected, so is
$\widetilde{\Omega }$; (ii) Among all the $\widetilde{\Omega }$'s above,
there exists a unique domain, denoted $\Omega ^f$, which is maximal
in the inclusion meaning.
\end{proposition}

\begin{proposition}[\cite{h1}] \label{prop2.4}
Let $A\subset \mathbb{C}^n$  be a
connected open set, $f$ and $g$  be two holomorphic functions
in $A$  with values in a complex Banach space $E$. If there
exists an open subset $U$ of $A$ such that $f(z)=g(z)$
for every $z$ in $U\cap \mathbb{R}^n$, then $f(z)=g(z)$
for every $z$ in $A$.
\end{proposition}


\begin{theorem} \label{thm2.5}
Let $D$ be a simply connected domain of
$\mathbb{R}^2\simeq \mathbb{C}$, with $D\neq \emptyset$, and
$\partial D\neq \emptyset $.  Let
$\mathcal{H}(D)=\{z\in \mathbb{C}^2;z_1+iz_2\in D
\mbox{ and }\bar{z}_1+i\bar{z}_2\in D\}$
be the harmonicity cell of $D$. Then

\noindent (1) For every $\zeta \in \partial \mathcal{H}(D)$,  and every open
neighbourhood $V_\zeta $ of $\zeta $ in $\mathbb{C}^2$,
there exists a classical ($\in C^2$) $\infty $-harmonic
function $f_\zeta $ on $D$, whose complex extension is
holomorphic in $\mathcal{H}(D)$,  but cannot be analytically continued
through $V_\zeta $.

\noindent(2) For the given domain $D$, let us denote by
$\mathcal{H}_\infty (D)$ the interior in $\mathbb{C}^2$
of $\cap \{D^u;u\in {\bf H}_\infty (D)\}$.
The set $\mathcal{H}_\infty (D)$ which may be called the
infinite-harmonicity cell of $D$, satisfies:
\begin{itemize}
\item[(a)] The trace of $\mathcal{H}_\infty (D)$ with $\mathbb{R}^2$ is $D$,
under the hypothesis that $\mathcal{H}_\infty (D)\neq \emptyset $

\item[(b)] $\mathcal{H}_\infty (D)$ is a connected open of
$\mathbb{C}^2$

\item[(c)] The inclusion $\mathcal{H}_\infty (D)\subset \mathcal{H}(D)$
always holds

\item[(d)] If $D$ is such that every $u\in {\bf H}_\infty (D)$ extends
holomorphically to $\mathcal{H}(D)$ then
$\mathcal{H}_\infty (D)\neq \emptyset$,
and both the cells $\mathcal{H}(D)$ and $\mathcal{H}_\infty (D)$ coincide.

\item[(e)] Suppose $D$ is bounded and covered by a finite union of open
rectangles $P_2^r(a_j;\rho _{j1},\rho _{j2})$, centered at $a_j\in D$,
$j=1,\dots,m$, such that for every $u\in {\bf H}_\infty (D)$
\[
\limsup_{n_k\to +\infty } \big[\frac 1{(n_k)!}
\big|\frac{\partial ^{n_k}u}{\partial x_k^{n_k}}(a_j)\big|\big]^{1/n_k}
\leq \frac 1{\rho _{jk}},\quad k=1,2,\; 1\leq j\leq m\,.
\]
Then $\mathcal{H}_\infty (D)\supset \cup_{j=1}^m  P_2^c(a_j,\rho _j)$,
and therefore $\mathcal{H}_\infty (D)\neq \emptyset $.
\end{itemize}
\end{theorem}

In the proof of Theorem \ref{thm2.5}, we will use the following two lemmas.

\begin{lemma} \label{lem2.6}
In every sector $-\pi <\theta <\pi $,  the
$\infty$-Laplace equation $\Delta _\infty u=0$  has a
solution in the form $u=\frac{v(\theta )}\rho $, where
$\theta=\mathop{\rm Arg}z$, $\rho =|z|$,  and $v$ satisfies the
ordinary differential equation (not containing $\theta $)
\begin{equation}
(v')^2v"+3v(v')^2+2v^3=0\label{e7}
\end{equation}
\end{lemma}

\begin{proof}
It is clear that we have to use polar coordinates. With
$x_1=\rho \cos \theta $, $x_2=\rho \sin \theta $ in \eqref{e6}, we get
by a simple calculation:
$u_{x_1}=u_\rho \cos \theta -\frac 1\rho u_\theta
\sin \theta $, $u_{x_2}=u_\rho \sin \theta +\frac 1\rho u_\theta \cos \theta
$, $u_{x_1x_1}=u_{\rho \rho }\cos ^2\theta +\frac 1{\rho ^2}u_{\theta \theta
}\sin ^2\theta -\frac 1\rho u_{\theta \rho }\sin 2\theta +\frac 1\rho u_\rho
\sin \theta +\frac 1{\rho ^2}u_\theta \sin 2\theta $, $u_{x_2x_2}=u_{\rho
\rho }\sin ^2\theta +\frac 1{\rho ^2}u_{\theta \theta }\cos ^2\theta +\frac
1\rho u_{\theta \rho }\sin 2\theta +\frac 1\rho u_\rho \cos ^2\theta -\frac
1{\rho ^2}u_\theta \sin 2\theta $, $u_{x_1x_2}=\frac 12u_{\rho \rho }\sin
2\theta -\frac 1{2\rho ^2}u_{\theta \theta }\sin 2\theta +\frac 1\rho
u_{\theta \rho }\cos 2\theta -\frac 1{2\rho }u_\rho \sin 2\theta -\frac
1{\rho ^2}u_\theta \cos 2\theta $. Finally, after expanding the terms and
rearranging, the $\infty $-Laplace equation \eqref{e6} takes the form (in polar
coordinates)
\begin{equation}
\Delta _\infty u=u_\rho ^2u_{\rho \rho }
+\frac{2u_\rho u_\theta u_{\rho\theta }}{\rho ^2}
+\frac{u_\theta ^2u_{\theta \theta }}{\rho ^4}
-\frac{u_\rho u_\theta ^2}{\rho ^3}=0 \label{e8}
\end{equation}
Putting $u=\frac{v(\theta )}\rho$ in \eqref{e8} we find
that $v$ satisfies the non-linear o.d.e. \eqref{e7}.
\end{proof}

\begin{lemma} \label{lem2.7}
Let  $D$ be a simply connected domain in $\mathbb{C}$, $D\neq \emptyset$,
$\partial D\neq \emptyset$.  For every $t\in \partial D$,  there exists
a complex valued $\infty$-harmonic function in $D$ which cannot be
extended continuously in any given open neighborhood of $t$.
\end{lemma}

\begin{proof}
Let us look for a solution of \eqref{e6} in $D$ in the form
$u(z)=\frac{v(\theta )}{|z-t|}$ , where the argument $\theta $ is the
unique angle in $]-\pi ,\pi [$ satisfying
$z-t=e^{i\theta }|z-t|,v$ is assumed to be $C^2$ in $]-\pi ,\pi [$.
Note here that the simple connexity of $D$ guarantees that $u$ is uniform
in $D$. As it can be shown
that the $\infty -$Laplacien operator:
$\Delta _\infty u=u_{x_1}^2u_{x_1x_1}+2u_{x_1}u_{x_2}u_{x_xx_2}
+u_{x_2}^2u_{x_2x_2}$
is invariant under translations $\tau _a$ of $\mathbb{C\simeq R}^2$,
$z=x_1+ix_2$, $a=a_1+ia_2$ - that is $\Delta _\infty (u\circ \tau
_a)=(\Delta _\infty u)\circ \tau _a$ - we may assume without loss of
generality that $t=0$. Insertion of $v=e^{\gamma \theta }$, where
$\gamma\in \mathbb{C}$ is a constant, in \eqref{e7} gives:
 $\gamma ^4+3\gamma ^2+2=0$ or $(\gamma ^2+1)(\gamma ^2+2)=0$. Take
$\gamma =i$ and consider the $\infty $-harmonic function in $D$ defined by:
$u(z)=\frac{e^{i\theta }}{|z-t|}$ , or more explicitly:
\[
u(z)=\begin{cases}
\frac 1{|z-t|}\exp (i\arcsin \frac{x_2-t_2}{|z-t|}) &\mbox{if }x_1\geq t_1 \\
\frac \pi {|z-t|}-\frac 1{|z-t|}\exp (i\arcsin \frac{x_2-t_2}{|z-t|}) &
\mbox{if  }x_1<t_1\text{ and }x_2>t_2 \\
\frac{-\pi }{|z-t|}-\frac 1{|z-t|}\exp(i\arcsin \frac{x_2-t_2}{|z-t|}) &
\mbox{if }x_1<t_1\text{ and } x_2<t_2
\end{cases}
\]
 The result follows immediately by taking the principal argument
$z\mapsto \mathop{\rm Arg}z\in ]-\pi ,\pi [$; here for
$\delta \in [-1,1]$, $\arcsin \delta $ signifies the unique number
$\beta $ in $[-\frac \pi 2,\frac \pi 2]$ satisfying $\sin \beta =\delta $.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2.5}]
For $\zeta \in \partial\mathcal{H}(D)$ there exists, due to
Proposition \ref{prop2.1}, a boundary point $t$ of $D$, such that
$\zeta \in \Gamma (t)$ (which is equivalent to $t\in T(\zeta )$).
As $T(w)$ reduces in the two-dimensional case to the pair
$\{w_1+iw_2,\quad \bar{w}_1+i\bar{w}_2\}$, we have $t=$
$\zeta _1+i\zeta _2$ or $t=\bar{\zeta}_1+i\bar{\zeta}_2$.

\noindent 1.a.\quad
Suppose at first that $t=\zeta _1+i\zeta _2$. By
Lemma \ref{lem2.6}, we deduce that  \eqref{e6} has a solution $u(z)$ in $D$
in the form $|z-t|^{-1}e^{i\theta }$. We conclude then as the
solutions of \eqref{e6} are in particular real analytic in $D$
(Proposition \ref{prop2.2}),
that the so-defined function $u(z)$ (given by Lemma \ref{lem2.7}) has a holomorphic
extension $\widetilde{u}$ to a maximal domain $A_1=D^u$ in $\mathbb{C}^2$
(Proposition \ref{prop2.3}). Since $\mathcal{H}(D)$ is the connected component
containing $D$ of the open
$\mathbb{C}^2-\cup_{t'\in \partial D}\{w\in \mathbb{C}^2;(w_1-t_1')^2
+(w_2-t_2')^2\}$, we have $A_1\supset \mathcal{H}(D)$. Substituting in $u(z)$
complex variables $w_1,w_2$ to real ones and putting
$h(w)=\sqrt{(w_1-t_1)^2+(w_2-t_2)^2}$, we obtain
\[
\widetilde{u}(w)=\begin{cases}
\frac 1{h(w)}\exp (i\arcsin \frac{w_2-t_2}{h(w)})
 &\mbox{if }\mathop{\rm Re} w_1\geq t_1 \\
\frac \pi {h(w)}-\frac 1{h(w)}\exp(i\arcsin \frac{w_2-t_2}{h(w)}(
&\mbox{if }\mathop{\rm Re}w_1<t_1\text{ and }\mathop{\rm Re}w_2>t_2 \\
\frac{-\pi }{h(w)}-\frac 1{h(w)}\exp(i\arcsin \frac{w_2-t_2}{h(w)})
&\mbox{if }\mathop{\rm Re}w_1<t_1\text{ and }\mathop{\rm Re}w_2<t_2,
\end{cases}
\]
where the branches are taken such that the square root is positive
when it is restricted to $D$, and for $\arcsin $ the branch is chosen such
that its values are real (in $]-\pi ,\pi [)$ whenever $z$ belongs to $D$. To
see that $\widetilde{u}(w)$ is holomorphic in $\mathcal{H}(D)$, we consider
\[
F(w)=\begin{cases}
\frac 1{g(w)}\exp(i\arcsin \frac{w_2-\mathop{\rm Im}(\zeta _1+i\zeta _2)}{g(w)})
&\mbox{ if }\mathop{\rm Re}w_1\geq \mathop{\rm Re}(\zeta _1+i\zeta _2) \\
\frac \pi {g(w)}-\frac 1{g(w)}\exp (i\arcsin \frac{w_2-\mathop{\rm Im}(\zeta
_1+i\zeta _2)}{g(w)})
&\mbox{if }\mathop{\rm Re}w_1<\mathop{\rm Re}(\zeta _1+i\zeta_2), \\
&\mathop{\rm Re}w_2>\mathop{\rm Im}(\zeta _1+i\zeta _2) \\[3pt]
\frac{-\pi }{g(w)}-\frac 1{g(w)}\exp(i\arcsin \frac{w_2-\mathop{\rm Im}(\zeta
_1+i\zeta _2)}{g(w)})
&\mbox{if }\mathop{\rm Re}w_1<\mathop{\rm Re}(\zeta _1+i\zeta _2),\\
&\mathop{\rm Re}w_2<\mathop{\rm Im}(\zeta _1+i\zeta _2),
\end{cases}
\]
where $g(w)=\sqrt{[(w_1+iw_2)-(\zeta _1+i\zeta _2)][(\bar{w}_1
+i\bar{w}_2)-(\zeta _1+i\zeta _2)]}$, and the branches are chosen as
in $\widetilde{u}(w)$. Seeing that by \cite{l1},
$\mathcal{H}(D)=\{w\in \mathbb{C}^2;T(w)\subset D\}$, and noting that $g(w)=0$
if and only if $w\in \Gamma(t) $ with $t\in \partial D$, the function $F(w)$
is well defined in some open $A_2\supset\mathcal{H}(D)$.
Observe that $\widetilde{u}$ and $F$ are both holomorphic in
$A=A_1\cap A_2$ - since $\frac{\partial \widetilde{u}}{\partial \bar{w}_j}
=\frac{\partial F}{\partial \bar{w}_j}=0$ in
$A\supset \mathcal{H}(D)$, $w_j=x_j+iy_j$ , $j=1,2$- having the same restriction
on $D=U\cap \mathbb{R}^2$, with $U=\mathcal{H}(D)$:
$\widetilde{u}|_D(z)=F|_D(z)=u(z)$, with $z=x_1+ix_2$.
By Proposition \ref{prop2.4}, we deduce
that $\widetilde{u}=F$ in $\mathcal{H}(D)$. Furthermore, since $\zeta $
and $t$ satisfies $(\zeta _1-t_1)^2+(\zeta _2-t_2)^2=0$, one has by letting
$w\in \mathcal{H}(D)$ tend to $\zeta$: $|\widetilde{u}(w)|=|h(w)^{-1}|\to \infty $;
consequently the function $\widetilde{u} (w)$ cannot be extended holomorphically
across $\zeta \in \partial \mathcal{H}(D)$.

\noindent 1.b\quad
If $t=\bar{\zeta}_1+i\bar{\zeta}_2$, the function
$G(w)$ defined in the same way by substituting $\bar{\zeta}_1+i
\bar{\zeta}_2$ to $\zeta _1+i\zeta _2$ in $F(w)$ (with similar
branches) satisfies: (i) $G(w)$ exists for every $w$ $\in \mathcal{H}(D)$, (ii)
$G(w)$ is holomorphic in $\mathcal{H}(D)$, (iii) $G(w)$ cannot be extended
holomorphically to any open neighborhood of $\zeta $ in $\mathbb{C}^2$ (since
$|G(w)|\to \infty $ when $w\in \mathcal{H}(D)\to \zeta $), (iv)
The restriction of $G(w)$ on $D$ is an infinite-harmonic function on $D$.

\noindent 2) It might happen that the set $\cap \{D^u;u\in {\bf H}_\infty (D)\}$
reduces to only the starting domain $D$, we would obtain thus an empty
$\infty $-harmonicity cell, and consequently (b), (c) are held to be true if
this eventual case occur.

\noindent (a) Suppose the above intersection is of non empty interior in
$\mathbb{C}^2$.
Since $D$ is considered as a relative domain in $\mathbb{R}^2$, with respect
to the induced topology from $\mathbb{C}^2$, and since
$D^u\cap \mathbb{R}^2=D $ for every $u\in {\bf \ H}_\infty (D)$, we have:
$\cap \{D^u;u\in {\bf H}_\infty (D)\}\cap \mathbb{R}^2
=\cap \{D^u\cap \mathbb{R}^2;u\in {\bf H}_\infty (D)\}=D;$ so
$\mathop{\rm Tr}\mathcal{H}_\infty (D)=\mathcal{H}_\infty (D)\cap
\mathbb{R}^2\subset D$. On the other hand, since $D\subset D^u $ for every
$u\in {\bf H}_\infty (D)$, we have
$D\subset (\cap_{u\in {\bf H}_\infty (D)} D^u)\cap \mathbb{R}^2$.
Moreover, from the real analyticity of a function $u\in $ ${\bf H}_\infty (D)$
in $D$, we deduce that for every point $a\in D$, there exist radius
$\rho _j^u=\rho_j^u(a)>0,j=1,2$, small enough such that
$u(z)=\sum_{\alpha \in \mathbb{N}^2} a_\alpha (z-a)^\alpha $, for all $z$
in the rectangle
$P_2^r(a,\rho _j^u(a))=\{x\in \mathbb{R}^2;|x_j-a_j|<\rho _j^u(a)$,
$j=1,2\}\subset D$, where
$(z-a)^\alpha =(x_1-a_1)^{\alpha_1}(x_2-a_2)^{\alpha _2}$.
Substituting $w\in \mathbb{C}^2$ to $z$, we obtain
$\widetilde{u}(w)=\sum_{\alpha \in \mathbb{N}^2} a_\alpha(w-a)^\alpha $ which
is of course holomorphic in the complex bidisk
$P_2^c(a,\rho _j^u(a))=\{w\in \mathbb{C}^2;|w_j-a_j|<\rho_j^u(a),j=1,2\}\subset
\mathbb{C}^2$, where $(w-a)^\alpha =(w_1-a_1)^{\alpha_1}(w_2-a_2)^{\alpha _2}$,
the chosen branch being such that the restriction
of $(w-a)^\alpha $ to $\mathbb{R}^2$ is $>0$. Thus the domain of holomorphic
extension of $u$ is nothing else but the union of all the
$P_2^c(a,\rho_j^u(a))$ 's with $a$ running through $D$. The above construction
involves $D\subset [\cap \{D^u;u\in {\bf H}_\infty (D)\}]^0
\cap \mathbb{R}^2$; so one has $\mathop{\rm Tr}\mathcal{H}_\infty (D)=D$.

\noindent (b) Let $w,w'$ be two arbitrary points in
$B=\cap \{D^u;u\in {\bf H}_\infty (D)\}$. By (a),
$B=\cap_{u\in {\bf H}_\infty (D)}\cup_{a\in D} P_2^c(a,\rho _j^u(a))$,
where $\rho _j^u(a)$, $j=1,2$, are the greatest radius corresponding to the
power series expansion  of $u$ at $a$. Note that the set $B$ is connected
in case the above intersection reduces to $D$. Suppose then $B\neq D$ and take
$w,w'$ in $B$. Since $w,w'$ are in $D^u$ for every
$u\in{\bf H}_\infty (D)$, there exist, by construction of $D^u$,
$a,a'\in D$, such that $w\in P_2^c(a,\rho _j^u(a))$, and
$w'\in P_2^c(a',\rho _j^u(a'))$. Putting
$\rho _j(a)=\inf \{\rho_j^u(a);u\in {\bf H}_\infty (D)\},\rho _j(a')=\inf \{\rho
_j^u(a');u\in {\bf H}_\infty (D)\}$, we obtain
$w\in P_2^c(a,\rho_j(a))$, $w'\in P_2^c(a',\rho _j(a'))$,
with $\rho _j(a)\geq 0$ and $\rho _j(a')\geq 0$. Let
then $\beta $ denote a path in $D$ joining
$\mathop{\rm Re}w\in P_2^r(a,\rho _j(a))\subset D$ to
$\mathop{\rm Re}w'\in P_2^r(a',\rho_j(a'))\subset D$.
The path $\gamma $, constituted successively
with the paths $[w,\mathop{\rm Re}w]$, $\beta $, and $[\mathop{\rm Re}w',w']$
joins $w$ to $w'$ and is included into the union
$P_2^c(a,\rho _j(a))\cup D\cup P_2^c(a',\rho _j(a'))\subset D^u$.
We conclude that $\gamma \subset B$, $B$ is connected and
therefore so is $\mathcal{H}_\infty (D)=B^0$.

\noindent (c) By contradiction, suppose that $\mathcal{H}(D)$ does not contain
$\mathcal{H}_\infty (D)$. Take $w_0\in \mathcal{H}_\infty (D)$ with
$w_0\notin \mathcal{H }(D)$. Since $\mathcal{H}_\infty (D)$ is connected and
$D\subset \mathcal{H}_\infty (D)$, there would exist a continuous path
$\gamma _{w_0,a}$ joining $w_0$ to some point $a\in D$, with
$\gamma _{w_0,a}\subset \mathcal{H}_\infty (D)$. Next, due to the inclusion
$D\subset \mathcal{H}(D)$, we ensure the
existence of a point $\zeta _0$ belonging to
$\gamma _{w_0,a}\cap \partial \mathcal{H}(D)$. Due to Part 1 above, to
the boundary point $\zeta _0$ of $\mathcal{H}(D)$ corresponds some function
$f_{\zeta _0}$ which is $\infty$-harmonic in $D$ and whose extension
$\widetilde{f_{\zeta _0}}$ in $\mathbb{C}^2 $ is a holomorphic function
in $\mathcal{H}(D)$ which can not be
holomorphically continued beyond $\zeta _0$. Now, the $\infty $-harmonicity
cell $\mathcal{H}_\infty (D)$ is characterized by: (i) Every
$u\in {\bf H}_\infty (D)$ is the restriction on $D$ of a holomorphic
function $\widetilde{u}:\mathcal{H}_\infty (D)\to \mathbb{C}$;
(ii) $\mathcal{H}_\infty (D)$ is the maximal domain of $\mathbb{C}^2$,
in the inclusion sense, whose trace on $\mathbb{R}^2$ is $D$, and satisfying (i).
Then $\widetilde{f_{\zeta _0}}$ is not holomorphic at $\zeta _0$ with
$\zeta _0$ inside $\mathcal{H}_\infty (D)$, which contradicts the property (i).
 Consequently, the inclusion $\mathcal{H}_\infty (D)\subset \mathcal{H}(D)$
always holds.

\noindent (d) By Proposition \ref{prop2.3}, given $u\in {\bf H}_\infty (D)$,
there exists a maximal domain $D^u\subset \mathbb{C}^2$ to which $u$ extends
holomorphically.
The domain $D^u$ is then a domain of holomorphy of $\widetilde{u}$ (also
called domain of holomorphy of $u$). Suppose that every $u\in {\bf H}_\infty (D)$
extends holomorphically to $\mathcal{H}(D)$. One has then
$\mathcal{H}(D)\subset D^u$, for every $u\in {\bf H}_\infty (D)$; therefore,
$\mathcal{H}(D)=\mathcal{H}(D)^0\subset [\cap_{u\in {\bf H}_\infty (D)}D^u]^0=
\mathcal{H}_\infty (D)$. The result follows by (c).

\noindent (e) Due to Proposition \ref{prop2.2}, every $\infty $-harmonic function
$u$ in $D$ is in particular real analytic in $D$, and thereby partially real
analytic in $D $. Since $D\subset \cup_{j=1}^m P_2^r(a_j,\rho_j)$, there exist
open rectangles $P_2^r(a_j,\rho _j^u)\subset D$ in which
$u$ writes as the sum of a power series in $(x_1-a_{j1})(x_2-a_{j2})$. More,
the convergence radius $\rho _{j1}^u,\rho _{j2}^u$ corresponding to the
development of $x_1\mapsto u(x_1,a_{j2})$ and $x_2\mapsto u(a_{j1},x_2)$
at $a_{j1}$ and $a_{j2}$ (respectively) are given by
$\rho _{jk}^u=\{\limsup_{n_k\to +\infty } [\frac 1{(n_k)!}|
\frac{\partial ^{n_k}u}{\partial x_k^{n_k}}(a_j)|]^{1/n_k}\}^{-1}$
$k=1,2$, $1\leq j\leq m$. By assumption, the given
covering of $D$ satisfies $\inf_{u\in {\bf H}_\infty (D)}\rho_{jk}^u\geq \rho _{jk}$,
 that is for every $x\in P_2^r(a_j,\rho _j)$:
\[
u(x)=\sum_{n_1\in \mathbb{N}} \sum_{n_2\in \mathbb{N}}
\frac 1{n_1!\; n_2!}\frac{\partial ^{n_1+n_2}u}
{\partial x_1^{n_1}\; \partial x_2^{n_2}}(a_j)
(x_1-a_{j1})^{n_1}(x_2-a_{j2})^{n_2},
\]
where $x=(x_1,x_2)$, $a_j=(a_{j1},a_{j2})$ and $\rho _j=(\rho _{j1,}\rho _{j2})$.
It is clear that
the complex series obtained by substituting $w_1,w_2\in \mathbb{C}$ to
$x_1,x_2\in \mathbb{R}$ is convergent on every complex bidisk
$P_2^c(a_j,\rho_j)=\{w\in \mathbb{C}^2;|w_1-a_{j1}|<\rho _{j1}$ and
$|w_2-a_{j2}|<\rho _{j2}\}$. Due to the maximality of $D^u$, we have
$\cup_{j=1}^m P_2^c(a_j,\rho _j)\subset D^u$ for
every $u\in {\bf H}_\infty (D)$, and thereby
$\cup_{j=1}^m  P_2^c(a_j,\rho _j) \subset \cap \{D^u;u\in {\bf H}_\infty (D)\}$.
The last union being an open set, one
deduces that $\mathcal{H}_\infty (D)\supset \cup_{j=1}^m P_2^c(a_j,\rho _j)$;
this mean in particular that $\mathcal{H}_\infty (D)\neq \emptyset$.
\end{proof}

\begin{remark} \label{rmk2.8} \rm
 The significant fact of the inclusion $\mathcal{H}_\infty(D)\subset \mathcal{H}(D)$
is that the common complex domain $\widetilde{D}$,
denoted $\mathcal{H}_\infty (D)$, for the whole class ${\bf H}_\infty (D)$,
cannot pass beyond $\mathcal{H}(D)$. Nevertheless, given a specified $\infty -
$harmonic function $u$ in $D$, we may have:
$D^u \supset \mathcal{H}(D)$ with $D^u\neq \mathcal{H}(D)$\,.
\end{remark}

\begin{example} \label{ex2.9}\rm
 Consider $D=\{(x_1,x_2)\in \mathbb{R}^2;x_1>0,x_2>0\}$, and
look for a $C^2$ solution $u$ in $D$ of
$\Delta _\infty u=0$ in the form $u=Ax_1^\alpha +Bx_2^\beta $
(where $A,B,\alpha ,\beta $ are constant).
 Since $\Delta _\infty u=A^3\alpha ^3(\alpha -1)x_1^{3\alpha
-4}+B^3\beta ^3(\beta -1)x_2^{3\beta -4}$, we deduce that
$u=x_1^{\frac43}-x_2^{\frac 43}$ is a classical $\infty $-harmonic function $u$
in $D$. Putting $w_j=x_j+iy_j$, $j=1,2$ and
$\widetilde{u}(w_1,w_2)=w_1^{\frac43}-w_2^{\frac 43}$, where the branch is chosen
such that the restriction of
$\widetilde{u}$ to $D\subset \mathbb{R}^2$ is a real valued function, we
observe that $\widetilde{u}$ is holomorphic in $\mathbb{C}^2-(L_1\cup L_2)$,
where $L_1=\mathbb{C}\times \{0\}$, $L_2=\{0\}\times \mathbb{C}$, and
$\widetilde{u}|D=u$. Since $\mathbb{C}^2-(L_1\cup L_2)=\mathbb{C}^{*}\times
\mathbb{C}^{*}$ is a domain (connected open) in $\mathbb{C}^2$, we deduce that
$D^u=\mathbb{C}^{*}\times \mathbb{C}^{*}$. The harmonicity cell of $D$ is given
explicitly by the set of all $w\in \mathbb{C}^2$ satisfying:
$w_1+iw_2=x_1-y_2+i(x_2+y_1)\in D$ and $\bar{w}_1+i\bar{w}_2
=x_1+y_2+i(x_2-y_1)\in D$ (here $\mathbb{R}^2\simeq \mathbb{C}$ ).
Thus $\mathcal{H}(D)=\{w\in \mathbb{C}^2;$ $x_1>|y_2|$ and
$x_2>|y_1|\}\subset D^u$, and $\mathcal{H}(D)\neq D^u$.
\end{example}

\begin{remark} \label{rmk2.10}\rm
 The inclusion $\mathcal{H}_\infty (D)\subset \mathcal{H}(D)$
can be strengthened. Indeed, let $D\subset \mathbb{C}$ be a simply connected
domain, with smooth boundary, and let ${\bf H}_{qr}{\bf (}D{\bf )}$ denote
the sub-class of all $\infty $-harmonic functions which are quasi-radial
with respect to some boundary point of $D$. A function $u\in {\bf H}_{qr}
{\bf (}D{\bf )}$ if there exists $t\in \partial D$ such that $u(z)=\rho
^mf(\theta )$, where $z=t+\rho e^{i\theta }\in D$, $f$ is a real or
complex-valued $C^2$ function in $]-\pi ,\pi [$, and $m$ is a constant (no
restriction on $m$ also). Note that by Aronsson \cite{a2}, ${\bf H}_{qr}{\bf (}D
{\bf )}$ is not empty. For instance, for $m>1$, one can find functions
$Z=f(\theta )$ in parametric representation:
$Z=\frac Cm(1-\frac 1m\cos ^2\tau )^{\frac{m-1}2}\cos \tau$,
$\theta =\theta _0+\int_{\tau _0}^\tau \frac{\sin ^2\tau '}{m-\cos ^2\tau '}d\tau '$,
$\tau _1<\tau <\tau _2$ ($C$, $\theta _0$, $\tau _0$, $\tau _1$, $\tau _2$
are constants). Similarly, let $\mathcal{H}_{qr}(D)$ denote the complex
domain $\widetilde{D}$ corresponding to ${\bf H}_{qr}(D)$. Since
$\mathcal{H}_{qr}(D)=[\cap_{u\in {\bf H}qr(D)} D^u]^0$ ,
${\bf H}_{qr}{\bf (}D{\bf )\subset H}_\infty (D)$, and the constructed
function $f_\zeta $ in the proof of Theorem \ref{thm2.5} is quasi-radial, we have:
$\mathcal{H}_\infty (D)\subset \mathcal{H}_{qr}(D)\subset \mathcal{H}(D)$.
\end{remark}

\begin{remark} \label{rmk2.11} \rm
To see that the property: $\lim_{w\to \zeta}|\widetilde{f}(w)|=\infty$,
($w\in \mathcal{H}(D)$, $\zeta \in \partial \mathcal{H}(D)$) may fail,
we give the following example.
\end{remark}

\begin{example} \label{ex2.12}\rm
 Let $D$ be an arbitrary simply connected plane domain,
$D\neq \emptyset$, $\partial D\neq \emptyset$. For a fixed
$\zeta \in \partial\mathcal{H}(D)$, take $t=\zeta _1+i\zeta _2\in T(\zeta )$
and consider
\[
F(w)=\sqrt{(w_1-t_1)^2+(w_2-t_2)^2}\exp (\frac 12\arctan\frac{w_2-t_2}{w_1-t_1}),
\]
where the branches are taken such that their restriction to
$D\subset \mathbb{R}^2$ is positive for the square root and in
$]-\frac \pi 2,\frac \pi 2[$ for $arctg)$. This function verifies: $F(w)$
is well defined and holomorphic on $\mathcal{H}(D)$, its restriction $f$ to $D$
is $\infty$-harmonic in $D$ since $f(z)=\sqrt{\rho }e^{\theta /2}$
where $z-t=\rho e^{i\theta }$; nevertheless $\lim_{w\to \zeta }|F(w)|=0$.
Indeed, if $\zeta $ is assumed in $\partial \mathcal{H}(D)-\partial D$,
one has $w_1+iw_2\to \zeta _1+i\zeta _2$, so that
$(w_1-t_1)^2+(w_2-t_2)^2=[(w_1+iw_2)-(\zeta _1+i\zeta _2)][(\bar{w}_1+i
\bar{w}_2)-(\zeta _1+i\zeta _2)]$ $\to 0$;
on the other hand, by definition of
$T(\zeta )$, $(\zeta _1-t_1)^2+(\zeta _2-t_2)^2=0$, thus
$\arctan\frac{w_2-t_2}{w_1-t_1}\to \arctan\frac{\zeta _2-t_2}{\zeta _1-t_1
}=\arctan\pm i=\pm i\infty $, and
$|\exp (\frac 12arctg\frac{w_2-t_2}{w_1-t_1})|\to 1$. Otherwise, the result
is immediate if $\zeta \in \partial D\subset \partial \mathcal{H}(D)$.
\end{example}

section{Holomorphic extension in Fluids dynamic} %sec 3

In this section, we consider two general examples where the above techniques,
of complexification and analytic continuation to $\mathbb{C}^n$, are used
for  the study of some physical problems. The main application we are
interested in is the problem of the behaviour of a flow near an extreme
point. In the following, ${\bf H}_p(D)$ denotes the class of all $p$-harmonic
functions on $D$.

\begin{proposition} \label{prop3.1}
Let $D\subset \mathbb{C}$ be an
arbitrary profile limited by a connected closed curve $C$,  and
consider a stationary plane flow round $D$ defined by the data of a
vanishing point and its  velocity $V_{\infty }$ at the
infinite. Suppose that $C$ contains two straight segments $[a,z_1]$,
$[a,z_2]$  originated at $a=a_1+ia_2$ and forming an angle
$\nu \pi ,0<\nu <1$. Then there exist a suitable real $p>1$ and
an open simply connected neighborhood $U$ of $a$,  such that
the quasi-linear p.d.e:
$\Delta _pu=|\nabla u|^2\Delta u+(p-2)\Delta _\infty u=0$,  has a radial
 (with respect to $a $) positive solution $\varphi $ in $U$,  which
approximates the modulus of the velocity $V(z)$ of the fluid. More
precisely:
\begin{itemize}
\item[(i)] $|V(z)|\sim \varphi (z)$ as $z\to a$,
($z\in U$).

\item[(ii)] $\varphi \in {\bf H}_{(3\nu -4)/(2\nu -2)}(U)$.

\item[(iii)] Let $C>0$ be a constant, and put
$\delta =(\frac{2-\nu }\nu C)^{(\nu -2)/(2\nu -2)}$; then a stream function
$\varphi _c$ associated with a function $\varphi $ of the form
$C|z-a|^{\nu/(2-\nu)}$  is given by
\[
\varphi _c(x_1+ix_2)=\begin{cases}
\delta \arcsin \frac{x_2-a_2}{|z-a|} &\mbox{if } x_1\geq a_1 \\
\delta \pi -\delta \arcsin \frac{x_2-a_2}{|z-a|} &\mbox{if } x_1<a_1
\text{ and }x_2>a_2 \\
-\delta \pi -\delta \arcsin \frac{x_2-a_2}{|z-a|} &\mbox{if } x_1<a_1
\text{ and }x_2<a_2
\end{cases}
\]
which is $q$-harmonic in $U$,  with $\frac 1p+\frac 1q=1$,;  that is,
$\varphi _c\in {\bf H}_{(3\nu -4)/(\nu -2)}(U)$.
\end{itemize}
\end{proposition}

\begin{proof} Since $D$ is connected and simply connected, there exists an
injective holomorphic transformation $f_1$ sending the closed lower
half-plane
\[
P^{-}-\{-i\}=\{\beta \in \mathbb{C}:\mathop{\rm Im}\beta \leq 0,\beta
\neq -i\},
\]
sharpened at $-i$, onto the the exterior of $D$, with
$a=f_1(\beta _0)$ and $\mathop{\rm Im}\beta _0=0$. The composed map
$\gamma=g_1(\beta )=[f_1(\beta )-a]^{1/(2-\nu)}$ is holomorphic,
injective, and sends an open simply connected neighborhood
$\mathcal{V}(\beta_0)\subset P^{-}-\{-i\}$ of $\beta _0$ onto a neighborhood
$\mathcal{V}(0)\subset P_1$, where $P_1\subset \mathbb{C[\gamma ]}$ is one
of the half-planes limited by the straight line passing through
$\gamma _1=(z_1-a)^{1/(2-\nu)}$ and $\gamma _2=(z_2-a)^{1/(2-\nu)}$.
Due to the symmetry principle of Schwartz \cite{h1}, the function $g_1$
extends as a holomorphic function $\widetilde{g_1}$ in some open simply connected
neighborhood $\widetilde{\mathcal{V}}(\beta _0)\subset \mathbb{C}-\{-i\}$,
$\widetilde{\mathcal{V}}(\beta _0)\supset $ $\mathcal{V}(\beta _0)$. Thus, for
every $\beta \in $ $\widetilde{\mathcal{V}}(\beta _0)$, one has the absolutely
convergent expansion for $\widetilde{g_1}: \widetilde{g_1}(\beta )=
\sum_{j=1}^{+\infty } A_j(\beta -\beta _0)^j$.
Moreover, seeing that $\widetilde{g_1}$ is holomorphic and injective in a
neighborhood of $\beta _0$, the first coefficient
$A_1=[\widetilde{g_1}]'(\beta _0)=g_1'(\beta _0)$ is $\neq 0$. This implies in
particular:
$ f_1(\beta )=a+(\beta -\beta _0)^{2-\nu }f_2(\beta )$, for
$\beta \in \mathcal{V}(\beta _0)$, where the function
$f_2(\beta )=[\sum_{j=1}^{+\infty } A_j(\beta -\beta _0)^{j-1}]^{2-\nu }$ is a
holomorphic function in $\mathcal{V}(\beta _0)$ and may be taken uniform
(seeing that $A_1$ is $\neq 0$) and holomorphic in a certain open
neighborhood $\widetilde{\mathcal{V}}_1(\beta _0)$. Thus,
for $\beta \in \mathcal{V}_1(\beta _0)$, one has
$f_1(\beta )=a+(\beta -\beta _0)^{2-\nu }
\sum_{j=0}^{+\infty } B_j(\beta -\beta _0)^j$, and
\[
f_1(\beta )-a \sim B_0(\beta-\beta _0)^{2-\nu }\quad \mbox{as }\beta \to \beta _0
\,\label{e9}
\]
where $B_0=A_1^{2-\nu }$. Recall that the flow is supposed to be
held round a profile $D$ with an angular point $a$. Due to the well known
Chaplygine condition, the vanishing point of the current moves under the
effect of viscosity and the formation of whirlpools, to the extreme point $a$
of $\bar{D}$. As a simple calculus reveals, the complex potential of
the flow round $D$ is given by:
\[
w=f(z)=Re^{-i\theta }g(z)+\frac{Re^{i\theta }r^2}{g(z)}
-[2irR\sin (\psi -\theta )]\ln (z),
\]
where $\mu =g(z)$ is the bijective holomorphic function from $D^c$,
the exterior of $D$, onto the domain $|\mu |>r$. The values of $r$ and $\psi $
are such that
$\lim_{z\to \infty } g(z)=\infty $,
$\lim_{z\to \infty } g'(z)=1$, $\mu _0=g(a)=re^{i\psi }$ and
$V_\infty =Re^{i\theta }$ is the velocity at the infinite. The
holomorphic bijection $f_3=f_1^{-1}\circ g^{-1}$ maps
$\{|\mu |\geq r\}$ onto $P^{-}-\{-i\}$. Thus \eqref{e9} gives
\begin{equation}
f_1\circ f_3(\mu )-f_1\circ f_3(\mu _0)
\sim B_0[f_3(\mu )-f_3(\mu _0)]^{2-\nu }\quad \mbox{as }
\mu \to \mu _0\,. \label{e10}
\end{equation}
 Since $f_3'(\mu _0)\neq 0$ one has
 $f_3(\mu)-f_3(\mu _0) \sim f_3'(\mu _0)(\mu -\mu _0)$ as
$\mu \to \mu _0$, so that \eqref{e10} implies
$g^{-1}(\mu )-g^{-1}(\mu _0)\sim C_0(\mu -\mu _0)^{2-\nu }$, where
$$
C_0=B_0 f_3'(\mu _0)^{2-\nu }=[\frac{g_1'(\beta_0).g'(a)}{f_1'(\beta _0)}]^{2-\nu };
$$
that is, $g(z)-g(a)\sim C_0^{1/(\nu -2)}(z-a)^{1/(2-\nu)}$ as $z\to a$.
Consequently, near the vanishing point $a$ of the flow, the derivative
of $g$ satisfies
\begin{equation}
g'(z) \sim \frac{g(z)-g(a)}{z-a}
\sim C_0^{1/(\nu-2)}(z-a)^{1/(2-\nu)-1}
=C_0^{1/(\nu -2)}(z-a)^{(\nu -1)/(2-\nu)} \label{e11}
\end{equation}
as $z\to a$.
On the other hand, putting $\mu =g(z)$, we obtain
\begin{equation}
\frac{df}{d\mu }=R\text{ }e^{-i\theta }-R\text{ }e^{i\theta }\frac{r^2}{\mu
^2}-\frac{2irR}\mu \sin (\psi -\theta )  \label{e12}
\end{equation}
Since the velocity satisfies $V(z)=\bar{f'(z)}$, for
$z\in D^c$, Equality \eqref{e12} at $\mu _0$ gives
\[
Re^{-i\theta }-Re^{i\theta }\frac{r^2}{\mu _0^2}-\frac{2irR}{\mu _0}\sin
(\psi -\theta )=0  %\eqref{e13}
\]
 From the above equation and \eqref{e12}, we get for
$|\mu |\geq r:\frac{df}{d\mu }(\mu )-\frac{df}{d\mu }(\mu _0)
=(\mu -\mu _0)h(\mu )$, with
$h(\mu )=r^2R$ $e^{i\theta }\frac{\mu +\mu _0}{\mu _{}^2\mu _0^2}
+\frac{2irR}{\mu\mu _0}\sin (\psi -\theta )$.
By a simple calculus, $\lim_{\mu\to \mu _0}
h(\mu )=\frac{2R}re^{-2i\psi }\cos (\theta -\psi)\neq 0$,
here we will have to suppose that $V_\infty $ is such that
$\theta \neq \psi \pm \frac \pi 2$ (otherwise, if $\theta =\psi \pm \frac \pi 2$,
a direct calculus will do). Hence,
\begin{equation}
\frac{df}{d\mu } \sim D_0(z-a)^{1/(2-\nu)} \quad \mbox{as } \mu \to \mu _0\,,
 \label{e14}
\end{equation}
 with $D_0=2C_0^{1/(\nu -2)}R\cos (\theta -\psi )/(re^{2i\psi })$.
 Writing $\frac{df}{dz}=\frac{df}{d\mu }.\frac{d\mu }{dz}$
and combining \eqref{e11} and \eqref{e14}, we obtain the equivalence
$\frac{df}{dz}\sim C_0^{1/(\nu-2)}D_0(z-a)^{1/(2-\nu)}(z-a)^{\frac{\nu -1}{2-\nu }}$
as $z\to a$.
Consequently $|V(z)| \sim  C|z-a|^{\nu/(2-\nu)}$, where
$$
C=\frac{2R|\cos (\theta -\psi )|}r|\frac{f_1'(\beta _0)}{g_1'(\beta _0).g'a)}|.
$$
Therefore, (i) and (ii) may be obtained by taking
$p=\frac{3\nu -4}{2\nu -2}$, $\eta =0$, $\varepsilon =\frac{\nu C}{2-\nu }$
in the following lemma.
\end{proof}

\begin{lemma} \label{lem3.2}
For every real $p>1$ and fixed complex point
$z_0\in \mathbb{C}$,  the $p$-Laplace equation $\eqref{e3}$ has
radial solutions (with respect to the origin point $z_0$)
defined in any sharpened disk $X^{*}$ at $z_0$:
$X^{*}=\{z\in \mathbb{C}$; $0<|z-z_0|<R_0\}$.
 All these functions may be given by:
 $\varepsilon \frac{p-1}{p-2}|z-z_0|^{\frac{p-2}{p-1}}+\eta $, if
$p\neq 2$, and $\varepsilon \ln |z-z_0|+\eta $, if
$p=2$,  where $a,b$ are arbitrary in $\mathbb{R}$.
\end{lemma}

\begin{proof}[Proof of Lemma \ref{lem3.2}]
 Since $\Delta _p(u\circ \tau _{z_0})=(\Delta_pu)\circ \tau _{z_0}$,
 we may assume that $z_0=0$. Firstly, the case $p=2$
is well known since a $2$-harmonic function $u$ in a domain $\Omega $ of
$\mathbb{R}^n$ is also harmonic outside the zeros of $grad$ $u$.
If $p\neq 2$ and if $x=\rho \cos \theta $, $y=\rho \sin \theta $ is used
in the $p$-Laplace equation
\begin{equation}
\Delta_pu=(p-2)[u_x^2u_{xx}+2u_xu_yu_{xy}+u_y^2u_{yy}]+(u_x^2+u_y^2)(u_{xx}+u_{yy})=0
\label{e15}
\end{equation}
we observe, via a simple substitution of $u_x$, $u_y$, $u_{xx}$,
$u_{yy}$, $u_{xy}$, expressed by means of the polar coordinates $(\rho $,
$\theta )$, and taking into account that the usual Laplace operator $\Delta $,
and the gradient of $u$ give in polar form:
$\Delta u=u_{\rho \rho}+\rho ^{-1}u_\rho +\rho ^{-2}u_{\theta \theta }$ ,
$|\nabla u|^2=u_\rho ^2+\rho ^{-2}u_\theta ^2$,
that \eqref{e15} takes the form
\begin{equation}
\Delta _pu=(p-2)[u_\rho ^2u_{\rho \rho }+\frac{2u_\rho u_\theta
u_{\rho \theta }}{\rho ^2}+\frac{u_\theta ^2u_{\theta \theta }}{\rho ^4}-
\frac{u_\rho u_\theta ^2}{\rho ^3}]+(u_\rho ^2+\frac{u_\theta ^2}{\rho ^2}
)[u_{\rho \rho }+\frac{u_\rho }{\rho }+\frac{u_{\theta \theta }}{\rho ^2}
]=0 \label{e16}
\end{equation}
To look for a radial solution $u$ of \eqref{e15}, it suffices to put
$u(x+iy)=h(\rho )$ in \eqref{e16}. We obtain
$\Delta _pu=(h')^2[(p-1)h''+\frac 1\rho h']=0$
which is computed without difficulty and gives the  result stated
in Lemma \ref{lem3.2}.

\noindent (iii) Writing \eqref{e15} in the divergence form, and seing that
$U$ is simply connected, one can associate to
$\varphi =C|z-a|^{\nu/(2-\nu)}$ a conjugate $q$-harmonic function $\varphi _c$
in $U$, defined by
$(\varphi _c)_{x_1}=-|\nabla \varphi |^{\nu /(2-\nu)}\varphi _{x_2}$ and
$(\varphi _c)_{x_2}=|\nabla \varphi |^{\nu /(2-\nu)}\varphi _{x_1}$.
\end{proof}

\begin{remark} \label{rmk3.3}\rm
There is a physical interpretation of the $p$-Laplace
equation \eqref{e3} in terms of the laminar pipe flow of so-called power-law fluids
\cite{a1}. Using the terminology of non-linear fluid mechanic, one is motivated
to call the stream function $v$, corresponding to the potential $u$, the
solution of  $\Delta _q v=\mathop{\rm div}(|\nabla v|^{q-2}\Delta v)=0$,
$1<q<+\infty $, where $\frac 1p+\frac 1q=1$. In the language of Potential
theory we say that $u$ and $v$ are conjugate functions.
\end{remark}

\begin{proposition} \label{prop3.4}
Under the same hypothesis than proposition above, suppose that $a$
 is a non angular point, $V(a+i\gamma )\neq 0$ for $\gamma $ real $\neq 0$
sufficiently small, and $\frac{\partial ^rV}{\partial x_2^r}(a)\neq 0$
 for some integer $r\geq 1$. Then in some neighborhood $U'$ of $a$, the
velocity of the fluid writes as
\begin{equation}
V(z)=[(x_2-a_2)^r+(x_2-a_2)^{r-1}h_1(x_1)+\dots+h_r(x_1)]h(z)=W(x_2-a_2)h(z)
\label{e17}
\end{equation}
where $z=x_1+ix_2$, $a=a_1+ia_2$, $h$ is
a real analytic function in some neighborhood $U'$ of
$a$ with $h(z)\neq 0$ for every $z\in U'$,  and
$h_1,\dots,h_r$, appearing in the  Weierstrass' unitary polynomial in
$(x_2-a_2)$,  are real analytic functions in some interval
$]a_1-\varepsilon ,a_1+\varepsilon [,\varepsilon >0$.
\end{proposition}

\begin{proof}
Due to Propositions \ref{prop2.3} and \ref{prop2.4} above,
we can extend holomorphically in $\mathbb{C}^2$ the velocity function
$V:\Omega =(\bar{D})^c\to \mathbb{C}$, $(x_1,x_2)\mapsto V(x_1+ix_2)$,
which is real analytic (in fact even antiholomorphic) in $\Omega $.
Using the same technique above, and putting:
$w=(w_1,w_2)=(x_1+iy_1,x_2+iy_2)\in \mathbb{C}^2$, we find a maximal domain
$\Omega ^V$ in $\mathbb{C}^2$ whose trace with
$\mathbb{R}^2$ is $\Omega $, and to which $V$ extends holomorphically. Let then
$\widetilde{V}$ denote the unique complexified function of $V$ with
$\widetilde{V}|_\Omega =V$ and $\widetilde{V}$ is holomorphic in
$\Omega^V$. Since $\widetilde{V}:\Omega ^V\to \mathbb{C}$ satisfies
also $\widetilde{V}(a)=V(a)=0$, $\widetilde{V}(a_1,a_2+\gamma)=V(a_1,a_2+\gamma )$
is $\neq 0$ for some $(a_1,a_2+\gamma )\in \Omega \subset \Omega ^V$ with
$\gamma \neq 0$, and $\frac{\partial ^r\widetilde{V}}{\partial w_2^r}(a)\neq 0$
-seing that
$\frac{\partial ^r\widetilde{V}}{\partial w_2^r}(a)
=\frac{\partial ^r\widetilde{V}}{\partial w_2^r}|_\Omega (a)
=\frac{\partial ^rV}{\partial x_2^r}(a)$ - there
exist, owing to Weierstass' preparation Theorem in $\mathbb{C}^n$ \cite[p.290]{r1}
with $n=2$, $r$ functions $H_1(w_1),\dots,H_r(w_1)$ which are holomorphic in
some open neighborhood $\widetilde{\Omega }_1$ of $a_1$ in $\mathbb{C}$, and a
function $H(w)$ which is holomorphic in some open neighborhood
$\widetilde{\Omega }\subset \Omega^V$ of $a$ in $\mathbb{C}^2$ with
$H(w)\neq 0$ in $\widetilde{\Omega }$, such that
\begin{equation}
\widetilde{V}(w)=[(w_2-a_2)^r+(w_2-a_2)^{r-1}H_1(w_1)+\dots+H_r(w_1)]H(w)
\label{e18}
\end{equation}
for every $w$ in some open neighborhood $(\widetilde{\Omega })'$
of $a$ in $\mathbb{C}^2$ with $(\widetilde{\Omega })'\subset $
$\widetilde{\Omega }\subset \Omega^V$. Taking now the restriction of
Equality \eqref{e18} to $\mathbb{R}^2$, and seeing that the restriction
$h_1,\dots,h_r $ of each holomorphic function $H_1(w_1),\dots,H_r(w_1)$ is (real)
analytic in $\widetilde{\Omega }_1\cap \mathbb{R}$, we find the announced
result \eqref{e17} by putting $H_j|_{\mathbb{R}^2}=h_j$,
$H|_{\mathbb{R}^2}=h$, and
$(\widetilde{\Omega })'\cap \mathbb{R}^2=U\subset \Omega $. Note also that
the restriction $h$ is analytic in $U$.
\end{proof}

Some concrete examples and physical interpretations of the above results
will be discussed in a further paper; nevertheless, the determination of the
$h_j$ 's rests heavily upon an identification process and a residue formula.
These functions stand for the analytic coefficients of what we will call the
Weierstrass polynomial associated to the velocity of the flow in
a neighborhood of a vanishing point.

Following Lelong's method who introduced the transformation $T$ in 1954
(which was useful for constructing the harmonicity cells defined by Aronszajn
in 1936), it seems advisable now that an analogue $T_\infty $ of $T$ must be
precise in order to give explicitly some infinite-harmonicity cells.

\begin{thebibliography}{00}

\bibitem{a1}  G. Aronsson, P. Lindqvist; \emph{On $p$-harmonic functions in the
plane and their stream functions}, J. Differential Equations 74.1, 157-178 (1988).

\bibitem{a2} G. Aronsson; \emph{On certain singular solutions of the
partial differential equation $u_x^2u_{xx}+2u_xu_yu_{xy}+u_y^2u_{yy}=0$},
manuscripta math. 47, p. 133-151 (1984).

\bibitem{a3}  N. Aronszajn; \emph{Sur les d\'{e}compositions des
fonctions analytiques uniformes et sur leurs applications}, Acta. math. 65
1-156 (1935).

\bibitem{a4}   V. Avanissian; \emph{Cellule d'harmonicit\'{e} et
prolongement analytique complexe}, Travaux en cours, Hermann, Paris (1985).

\bibitem{b1}  \ L. Bers; \emph{Mathematical Aspects of Subsonic and Transonic Gaz
Dynamics}. Surveys in Applied Mathematics III.\quad Wiley New-York (1958).

\bibitem{b2}   M. Boutaleb; \emph{Sur la cellule d'harmonicit\'{e} de la
boule unit\'{e} de $\mathbb{R}^n$}, Publications de L'I.R.M.A. Doctorat de 3$^0
$cycle, U.L.P. Strasbourg France (1983).

\bibitem{b3}  M. Boutaleb; \emph{A polyharmonic analogue of a Lelong theorem and
polyhedric harmonicity cells}, Electron. J. Diff. Eqns, Conf. 09 (2002),
 pp. 77-92.

\bibitem{b4}  M. Boutaleb;
\emph{G\'{e}n\'{e}ralisation \`{a} $\mathbb{C}^n$ d'un
th\'{e}or\`{e}me de M. Jarnicki sur les cellules d'harmonicit\'{e}}, \`{a}
para\^{\i}tre dans un volume of the Bulletin of Belgian Mathematical Society
Simon Stevin (2003).

\bibitem{b5}  M. Boutaleb; \emph{On the holomorphic extension for $p$-harmonic
functions in plane domains}, submitted for publication (2003).

\bibitem{b6}  M. Boutaleb; \emph{On polyhedric, and topologically homeomorphic,
harmonicity cells in $\mathbb{C}^n$}, submitted for publication (2003).

\bibitem{c1}   R. Coquereaux , A. Jadczyk; \emph{Conformal Theories,
Curved phase spaces, Relativistic wavelets and the Geometry of complex
domains}, Centre de physique th\'{e}orique, Section 2, Case 907. Luminy,
13288. Marseille. France.(1990).

\bibitem{e1}  P. Ebenfelt; \emph{Holomorphic extension of solutions of elliptic
partial differential equations and a complex Huygens' principle}, Trita Math
0023, Royal Institute of Technology, S-100 44 Stockholm, Sweden (1994).

\bibitem{h1}   M. Herv\'{e};
\emph{Les fonctions analytiques Presses Universitaires
de France}, 1982.

\bibitem{j1}  M. Jarnicki; \emph{Analytic Continuation of harmonic functions},
Zesz. Nauk.U J, Pr. Mat 17, 93-104 (1975).

\bibitem{k1}   C. O. Kiselman; \emph{Prolongement des solutions d'une
\'{e}quation aux d\'{e}riv\'{e}es partielles \`{a} coefficients constants},
Bull. Soc. Math. France 97 (4) 328-356 (1969).

\bibitem{l1}   P. Lelong; \emph{Prolongement analytique et
singularit\'{e}s complexes des fonctions harmoniques}, Bull. Soc. Math.
Belg.710-23 (1954-55).

\bibitem{l2}   J. L. Lewis; \emph{Regularity of the derivatives of solutions to
certain degenerate elliptic equations}, Indiana Univ. Math. J. 32,
849-858 (1983).

\bibitem{m1}  \ Z. Moudam; \emph{Existence et R\'{e}gularit\'{e} des solutions du
$p$-Laplacien avac poids dans $\mathbb{R}^n$}, D. E. S. Univ. S. M. Ben
Abdellah, Facult\'{e} des Sciences Dhar Mehraz (1996).

\bibitem{o1}  E. Onofri; \emph{SO(n,2)- Singular orbits and their quantization}.
Colloques Internationaux C.N.R.S. No. 237.
G\'{e}om\'{e}trie symplectique et Physique math\'{e}matique. Instituto di
Fisica de l'Universita. Parma. Italia (1976).

\bibitem{p1}  M. Pauri; \emph{Invariant Localization and Mass-spin relations in
the Hamiltonian formulation of Classical Relativistic Dynamics}, University of
Parma, IFPR-T 019, (1971).

\bibitem{r1}  W. Rudin; \emph{Function theory in the unit ball of
$\mathbb{C}^n$}, Springer-Verlag New -York Heidelberg Berlin (1980).

\bibitem{s1}  J. Siciak; \emph{Holomorphic continuation of harmonic
functions}, Ann. Pol. Math. XXIX 67-73 (1974).

\end{thebibliography}

\end{document}