The Hyperbolic Cauchy Problem
The approach to the Cauchy problem taken here by the authors is based on theuse of Fourier integral operators with a complex-valued phase function, which is a time function chosen suitably according to the geometry of the multiple characteristics. The cor
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Kunihiko Kajitani Tatsuo Nishitani
The Hyperbolic Cauchy Problem
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Authors Kunihiko Kajitani Institute of Mathematics University of Tsukuba Tsukuba 305, Japan Tatsuo Nishitani Department of Mathematics College of General Education Osaka University Toyonaka Osaka 560, Japan
Mathematics Subject Classification (1991): 35B99, 35Ll5, 35L25, 35L30
ISBN 3-540-55018-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-55018-6 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Printed in Germany
Typesetting: Camera ready by author Printing and binding: Druckhaus Beltz, HemsbachlBergstr. 46/3140-543210 - Printed on acid-free paper
Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
VB
Part I Fourier integral operators with complex-valued phase function and the Cauchy problem for hyperbolic operators - KUNIHIKO KAJITANI O. Preface...................................................................... 1 2 1. Introduction 5 2. Hyperbolic polynomials 3. Interpolation theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12 polynomials 16 4. 20 5. Almost analytic extension of symbols in Gevrey classes 25 6. Fourier integral operators with a complex-valued phase function 43 7. Wave front sets in Gevrey classes 8. Local existence and uniqueness theorems for the Cauchy problem. . . . . . . . . .. 46 58 9. Proofs of Theorem 1.1 and Theorem 1.2 62 10. Propagation of wave front sets in Gevrey classes 11. Proof of Theorem 1.3 67 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69
Part II The effectively hyperbolic Cauchy problem -
TATSUO NISHITANI
1. Effectively hyperbolic operators 71 71 1.1 Effective hyperbolicity, geometric characterization 74 1.2 Time function vanishing on the doubly characteristic set 2. Reduced forms of effectively hyperbolic operators. . . . . . . . . . . . . . . . . . . . . . . . . . .. 76 77 2.1 Reduced forms relative to time function 80 2.2 Sketch of deriving energy estimates 82 3. Localizations and related pseudo-differential operators 82 3.1 Related pseudo-differential operators 87 3.2 Localizations relative to time f
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