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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+\vert\alpha\vert\leq l+q$ appear. The problem is solved by the fixed-point theorem with appropriate estimations in a Banach space.

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Additional Information

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