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Singular Nonlinear Partial Differential Equations

Aspects Mathematics Edited by Klas Diederich Vol. E 3:

G. Hector/U. Hirsch: Introduction to the Geometry of Foliations, Part B

Vol. E 5:

P. Stiller: Automorphic Forms and the Picard Number of an Elliptic Surface

Vol. E 6:

G. Faltings/G. Wustholz et 01.: Rational Points*

Vol. E 7:

W. Stoll: Value Distribution Theory for Meromorphic Maps

Vol. E 9:

A. Howard/P.-M. Wong (Eds.): Contribution to Several Complex Variables

Vol. E 10: A. J. Tromba (Ed.): Seminar of New Results in Nonlinear Partial Differential Equations* Vol. E 15: J.-P. Serre: Lectures on the Mordell-Weil Theorem Vol. E 16: K. Iwasaki/H. Kimura/S. Shimomura/M. Yoshida: From Gauss to Painleve Vol. E 17: K. Diederich (Ed.): Complex Analysis Vol. E 18: W. W. J. Hulsbergen: Conjectures in Arithmetic Algebraic Geometry Vol. E 19:

R. Racke: Lectures on Nonlinear Evolution Equations

Vol. E 20:

F. Hirzebruch, Th. Berger, R. Jung: Manifolds and Modular Forms*

Vol. E 21:

H. Fujimoto: Value Distribution Theory of the Gauss Map of Minimal Surfaces in Rm

Vol. E 22:

D. V. Anosov/A. A. Bolibruch: The Riemann-Hilbert Problem

Vol. E 23: A. P. Fordy/J. C. Wood (Eds.): Harmonic Maps and Integrable Systems Vol. E 24:

D. S. Alexander: A History of Complex Dynamics

Vol. E 25: A. Tikhomirov/A. Tyurin (Eds.): Algebraic Geometry and its Applications Vol. E 26:

H. Skoda/J.-M. Trepreau (Eds.): Contributions to Complex Analysis and Analytic Geometry

Vol. E 27:

D. N. Akhiezer: lie Group Actions in Complex Analysis

Vol. E 28:

R. Gerard, H. Tahara: Singular Nonlinear Partial Differential Equations

* A Publication of the Max-Planck-Institut fur Mathematik, Bonn Volumes of the German-language subseries "Aspekte der Mathematik" are listed at the end of the book.

Raymond Gerard Hidetoshi Tahara

Singular Nonlinear

Partial Differential

Equations

II Vleweg

Professor Raymond Gerard Institut de Recherche MatMmatique Alsacien Universite Louis Pasteur 10 rue du General Zimmer F-67084 Strasbourg, France

Professor Hidetoshi Tahara Dept. of Mathematics Sophia University Kioicho, Chiyoda - ku 102 Tokyo, Japan

Mathematics Subject Classification: 47 E OS, 47 F OS, 34 A 20, 34 G 20, 35 A 10,35 F 20

All rights reserved © Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, 1996 Softcover reprint of the hardcover I st edition 1996 Vieweg is a subsidiary company of the Bertelsmann Professional Information.

No part of this publication may be reproduced, stored in a retrieval system or transmitted, mechanical, photocopying or otherwise without prior permission of the copyright holder.

Cover design: Wolfgang Nieger, Wiesbaden Printed on acid-free paper

ISSN 0179-2156 ISBN- 13: 978-3-322-80286-6 e-ISBN- 13: 978-3-322-80284-2 DOl: 10.1007/978-3-322-80284-2

v

Preface The aim of this book is to put together all the results that are known about the existence of formal, holomorphic and singular solutions of singular non linear partial differential equations. We study the existence of formal power series s