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Sur l’existence d’une solution ramifiée pour des équations de Fuchs à caractéristique simple

Author: Patrice Pongérard
Journal: Proc. Amer. Math. Soc. 137 (2009), 2671-2683
MSC (2000): Primary 35A07; Secondary 35A20
Published electronically: February 3, 2009
MathSciNet review: 2497480
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Abstract: The aim of this paper is to construct a holomorphic solution, ramified around a simple characteristic hypersurface, for some linear Fuchsian equation of order $m\geq 1$. We consider an operator $L$, holomorphic in a neighborhood of the origin in ${\mathbb {C}}_t\times {\mathbb {C}}_x^n$, of the form $L=tA+B$ where $A$ and $B$ are linear partial differential operators of order $m$ and $m-1$, and where $A$ has a simple characteristic hypersurface transverse to $S:t=0$. Under an assumption linking the principal symbols of $A$ and $B$, the question is reduced to the study of an integro-differential Fuchsian equation with an additional variable $z$ that describes the universal covering of a pointed disk. It is an equation where terms like $t^lD_t^hD_x^\alpha (tD_t+1)^{-1}D_z^{-q}, l,h,q\in \mathbb {N}, \alpha \in \mathbb {N}^n$ with $l\leq 1$ and $h+|\alpha |\leq l+q$ appear. The problem is solved by the fixed-point theorem with appropriate estimations in a Banach space.

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Patrice Pongérard
Affiliation: Université de La Réunion, 23 allée des rubis, 97400 Saint-Denis, La Réunion, France

Received by editor(s): February 25, 2008
Received by editor(s) in revised form: October 16, 2008
Published electronically: February 3, 2009
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2009 American Mathematical Society