Raymond Gerard Hidetoshi Tahara. Singular Nonlinear Partial Differential Equations

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1 Raymond Gerard Hidetoshi Tahara Singular Nonlinear Partial Differential Equations

2 Aspects Mathematics Edited by Klas Diederich Vol. E 3: Vol. E 5: Vol. E 6: Vol. E 7: Vol. E 9: Vol. E 10: G. Hector/U. Hirsch: Introduction to the Geometry of Foliations, Part B P. Stiller: Automorphic Forms and the Picard Number of an Elliptic Surface G. Faltings/G. Wustholz et 01.: Rational Points* W. Stoll: Value Distribution Theory for Meromorphic Maps A. Howard/P.-M. Wong (Eds.): Contribution to Several Complex Variables 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 R m 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.

3 Raymond Gerard Hidetoshi Tahara Singular Nonlinear Partial Differential Equations II Vleweg

4 Professor Raymond Gerard Institut de Recherche MatMmatique Alsacien Universite Louis Pasteur 10 rue du General Zimmer F 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 ISBN- 13: e-isbn- 13: DOl: /

5 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 solutions, holomorphic solutions, and singular solutions of singular non linear partial differential equations. In the first chapter, we introduce operators with regular singularities in the one variable case and we give a new simple proof of the classical Maillet's theorem for algebraic differential equations. In chapter 2, we extend this theory to operators in several variables. The chapter 3 is devoted to the study of formal and convergent power series solutions of a class of singular partial differential equations having a linear part, using the method of iteration and also Newton's method. As an application of the former results, we look in chapter 4 at the local theory of differential equations of the form xy' = 1(x,y) and, in particular, we show how easy it is to find the classical results on such an equation when 1(0,0) = 0 and give also the study of such an equation when 1(0,0) #- 0 which was never given before and can be extended to equations of the form Ty = F(x, y) where T is an arbitrary vector field. Using the classical method of majorant equations we are able in chapters 5 and 8 to give holomorphic and singular solutions of non linear singular partial differential equations. The chapters 6 and 7 are devoted to Maillet's type theorem for singular non linear partial differential equations, first with a linear part and then without linear part. In chapter 9, we give a very general result for the Cauchy problem for non linear partial differential equations. Finally, in chapter 10, we generalize Maillet's type theorems in chapters 6 and 7 to some non linear integro-differential equations. During the preparation of this book, the first author (R. G.) stayed twice in Japan and the second author (H. T.) stayed three times in Strasbourg. The first author thanks University of Tokyo and Kyushu University for supporting his stay in Japan. The second author thanks Universite Louis Pasteur for supporting his stay in Strasbourg. They also thank their many friends for interesting discussions on the thema of the book. The original manuscript of this book was written by a local word processor and O. Korten retyped it into the present form. The authors express their hearty gratitude to O. Korten for TEXing this book. Raymond Gerard Hidetoshi Tahara

6 VI Contents Preface V 1 Operators with regular singularities: One variable case 1 Introduction Notations, definitions, examples The good operators A class of operators with a regular singularity Applications to differential equations Non linear differential equations Some particular cases The Maillet theorem A generalization of theorem Application: The theorem of E. Maillet Some notes on the theorem of E. Maillet Operators with regular singularities: Several variables case 42 Introduction A Formal theory Notations Linear operators on q[xll Non linear operators on rrr f Solutions of linear equations Solutions of non linear equations Non linear equations with an affine part in (Op.l.f)b,t Non linear equations without affine part 54 B Analytic theory Notations and definitions The good operators and the notion of domination In (Op.l.f)t In (Op.l.f)b,t In (Op.l.f)b,d A class of operators having a regular singularity In (Op.l.f)d In (Op.l.f)b,d In (Op.l.f)b,t Applications to partial differential equations With a Poincare vector field The partial differential equations of the hypergeometric functions of P. Appell

7 VII The systems of partial differential equations of the confluent hyper geometric functions in two variables The systems of partial differential equations of the hypergeometric functions in n variables of G. Lauricella A theorem of M. Kashiwara, T. Kawai, and J. Sjostrand Formal and convergent solutions of singular partial differential equations 77 Introduction Notations and definitions Holomorphic solutions of certain equations Solution of the linear equation Solution of the non linear equation Equations with parameters An application: A theorem of S. Kaplan The case of small denominators Local study of differential equations of the form xy' = f(x, y) near x = Introduction Coupling of two differential equations Behavior of solutions of a differential equation near a regular point Local study of a differential equation near a singular point of regular type Singular point of regular type The singularities of solutions of the Briot-Bouquet equation Study of the Hukuhara equation and of the Hukuhara function Holomorphic and singular solutions of non linear singular first order partial differential equations 138 Introduction Notations and definitions Statement of the main theorem Holomorphic solutions Singular solutions Uniqueness of the solution Proof of the main theorem Remarks Supplementary result Maillet's type theorems for non linear singular partial differential equations 161 Introduction Implicit function theorem Non linear equations with first order linear part Non linear equations with higher order linear part 179

8 VIII 6.4 Formal Gevrey index for a particular type of equations - Examples Supplementary results Maillet's type theorems for non linear singular partial differential equations without linear part 187 Introduction Notations and definitions Assumptions and results A basic lemma Proof of theorem Complementary results A remark Holomorphic and singular solutions of non linear singular partial differential equations 203 Introduction Holomorphic solutions Singular solutions: Special case Singular solutions: General case Asymptotic study Completion of the proof of the main theorem On the existence of holomorphic solutions of the Cauchy problem for non linear partial differential equations 234 Introduction Notations and definitions Results Proof of theorem Proof of theorem Maillet's type theorems for non linear singular integro-differential e~d~~ 2~ Introduction Notations and definitions The main theorem Construction of the formal solution Some discussions Convergence of the formal solution in the case Sl = Convergence of the formal solution in the case Sl > Supplementary results and remark Bibliography 264 Index 268