\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: $$\Delta ^mu=\sum_{|\alpha |=m} \frac{m!}{\alpha !} \frac{\partial ^{2|\alpha |}u}{\partial x_1^{2\alpha _1}\dots\partial x_n^{2\alpha _n}}=0,\quad m=1,2,3\dots \label{e2}$$ 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 $D\subset \mathbb{R}^2$ \Delta _pu=\mathop{\rm div}(|\nabla u|^{p-2}\Delta u)=0,\quad 11$, the hodograph method transforms$\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 $$\mathcal{H}(\Omega )=\big\{z=x+iy\in \mathbb{C}^n: \max_{t\in T( iy)}{\max } \big[ \max_{\xi \in S^{n-1}}\big( \langle x+t,\xi \rangle- \max_{s\in \Omega }\langle \xi ,s\rangle \big) \big] <0\big\}$$ \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 $$[ \cap_{f\in {\bf Ha}(\Omega )} \Omega ^f]^0=\mathcal{H} (\Omega ), \label{e5}$$ 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$11$) 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)$$\limsup_{n_k\to +\infty } \big[\frac 1{(n_k)!} \big|\frac{\partial ^{n_k}u}{\partial x_k^{n_k}}(a_j)\big|\big]^{1/n_k} \leq \frac 1{\rho _{jk}},\quad k=1,2,\; 1\leq j\leq m\,.$ Then$\mathcal{H}_\infty (D)\supset \cup_{j=1}^m P_2^c(a_j,\rho _j)$, and therefore$\mathcal{H}_\infty (D)\neq \emptyset $. \end{itemize} \end{theorem} In the proof of Theorem \ref{thm2.5}, we will use the following two lemmas. \begin{lemma} \label{lem2.6} In every sector$-\pi <\theta <\pi $, the$\infty$-Laplace equation$\Delta _\infty u=0$has a solution in the form$u=\frac{v(\theta )}\rho $, where$\theta=\mathop{\rm Arg}z$,$\rho =|z|$, and$v$satisfies the ordinary differential equation (not containing$\theta $) $$(v')^2v"+3v(v')^2+2v^3=0\label{e7}$$ \end{lemma} \begin{proof} It is clear that we have to use polar coordinates. With$x_1=\rho \cos \theta $,$x_2=\rho \sin \theta $in \eqref{e6}, we get by a simple calculation:$u_{x_1}=u_\rho \cos \theta -\frac 1\rho u_\theta \sin \theta $,$u_{x_2}=u_\rho \sin \theta +\frac 1\rho u_\theta \cos \theta $,$u_{x_1x_1}=u_{\rho \rho }\cos ^2\theta +\frac 1{\rho ^2}u_{\theta \theta }\sin ^2\theta -\frac 1\rho u_{\theta \rho }\sin 2\theta +\frac 1\rho u_\rho \sin \theta +\frac 1{\rho ^2}u_\theta \sin 2\theta $,$u_{x_2x_2}=u_{\rho \rho }\sin ^2\theta +\frac 1{\rho ^2}u_{\theta \theta }\cos ^2\theta +\frac 1\rho u_{\theta \rho }\sin 2\theta +\frac 1\rho u_\rho \cos ^2\theta -\frac 1{\rho ^2}u_\theta \sin 2\theta $,$u_{x_1x_2}=\frac 12u_{\rho \rho }\sin 2\theta -\frac 1{2\rho ^2}u_{\theta \theta }\sin 2\theta +\frac 1\rho u_{\theta \rho }\cos 2\theta -\frac 1{2\rho }u_\rho \sin 2\theta -\frac 1{\rho ^2}u_\theta \cos 2\theta $. Finally, after expanding the terms and rearranging, the$\infty $-Laplace equation \eqref{e6} takes the form (in polar coordinates) $$\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}$$ 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_1t_2 \\ \frac{-\pi }{|z-t|}-\frac 1{|z-t|}\exp(i\arcsin \frac{x_2-t_2}{|z-t|}) & \mbox{if }x_1t_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\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_1a_2 \\ -\delta \pi -\delta \arcsin \frac{x_2-a_2}{|z-a|} &\mbox{if } x_1r. 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 $$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}$$ Since f_3'(\mu _0)\neq 0 one has f_3(\mu)-f_3(\mu _0) \sim f_3'(\mu _0)(\mu -\mu _0) as \mu \to \mu _0, so that \eqref{e10} implies g^{-1}(\mu )-g^{-1}(\mu _0)\sim C_0(\mu -\mu _0)^{2-\nu }, where C_0=B_0 f_3'(\mu _0)^{2-\nu }=[\frac{g_1'(\beta_0).g'(a)}{f_1'(\beta _0)}]^{2-\nu }; that is, g(z)-g(a)\sim C_0^{1/(\nu -2)}(z-a)^{1/(2-\nu)} as z\to a. Consequently, near the vanishing point a of the flow, the derivative of g satisfies $$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}$$ as z\to a. On the other hand, putting \mu =g(z), we obtain $$\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}$$ 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$h(\mu )=r^2Re^{i\theta }\frac{\mu +\mu _0}{\mu _{}^2\mu _0^2} +\frac{2irR}{\mu\mu _0}\sin (\psi -\theta )$. 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, $$\frac{df}{d\mu } \sim D_0(z-a)^{1/(2-\nu)} \quad \mbox{as } \mu \to \mu _0\,, \label{e14}$$ 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$z_0$:$X^{*}=\{z\in \mathbb{C}$;$0<|z-z_0|0$. \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 $$\widetilde{V}(w)=[(w_2-a_2)^r+(w_2-a_2)^{r-1}H_1(w_1)+\dots+H_r(w_1)]H(w) \label{e18}$$ 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. 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