2004-Fez conference on Differential Equations and Mechanics. Electron. J. Diff. Eqns., Conference 11, 2004, pp. 81-93.

An infinite-harmonic analogue of a Lelong theorem and infinite-harmonicity cells

Mohammed Boutaleb

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
$$ u_x^2u_{xx}+2u_xu_yu_{xy}+u_y^2u_{yy}=0\quad \hbox{in } D\subset \mathbb{R}^2\,, $$
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.

Published October 15, 2004.
Math Subject Classifications: 31A30, 31B30, 35J30.
Key Words: Infinite-harmonic functions; holomorphic extension; harmonicity cells; p-Laplace equation; stationary plane flow.

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Mohammed Boutaleb
Dép. de Mathématiques. Fac de Sciences Fès D. M
B.P. 1796 Atlas Maroc
email: mboutalebmoh@yahoo.fr

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