Hyperbolic Partial Differential Equations
Serge Alinhac (1948–) received his PhD from l'Université Paris-Sud XI (Orsay). After teaching at l'Université Paris Diderot VII and Purdue University, he has been a professor of mathematics at l'Université Pari
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Serge Alinhac
Hyperbolic Partial Differential Equations
123
Serge Alinhac Université Paris-Sud XI Département de Mathématiques Orsay Cedex 91405 France [email protected]
Editorial Board: Sheldon Axler, San Francisco State University Vincenzo Capasso, Università degli Studi di Milano Carles Casacuberta, Universitat de Barcelona Angus MacIntyre, Queen Mary, University of London Kenneth Ribet, University of California, Berkeley Claude Sabbah, CNRS, École Polytechnique Endre Süli, University of Oxford Wojbor Woyczyński, Case Western Reserve University
ISBN 978-0-387- 87822-5 e-ISBN 978-0-387-87823-2 DOI 10.1007/978-0-387-87823-2 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009928133 Mathematics Subject Classification (2000): 35Lxx © Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Contents Introduction 1 Vector Fields and Integral Curves
ix 1
1.1
First Definitions . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3
Directional Derivatives . . . . . . . . . . . . . . . . . . . . .
3
1.4
Level Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.5
Bracket of Two Fields . . . . . . . . . . . . . . . . . . . . .
5
1.6
Cauchy Problem and Method of Characteristics . . . . . . .
5
1.7
Stopping Time . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.8
Straightening Out of a Field . . . . . . . . . . . . . . . . . .
8
1.9
Propagation of Regularity . . . . . . . . . . . . . . . . . . .
9
1.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2 Operators and Systems in the Plane
13
2.1
Operators in the Plane: First Definitions . . . . . . . . . . .
13
2.2
Systems in the Plane: First Definitions . . . . . . . . . . . .
15
2.3
Reducing an Operator to a System . . . . . . . . . . . . . .
16
2.4
Gronwall Lemma . . . . . . . . . . . . . . . . . . . . . . . .
18
2.5
Domains of Determination I (A priori Estimate) . . . . . .
19
vi
Contents 2.6
Domains of Determination II (Existence)
. . . . . . . . . .
21
2.7
Exercises . . . . . . . . . . . . . .
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