Partial Differential Equations III Nonlinear Equations
The third of three volumes on partial differential equations, this is devoted to nonlinear PDE. It treats a number of equations of classical continuum mechanics, including relativistic versions, as well as various equations arising in differential geometr
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J.E. Marsden Control and Dynamical Systems, 107-81 California Institute of Technology Pasadena, CA 91125 USA [email protected]
L. Sirovich Laboratory of Applied Mathematics Department of Biomathematical Sciences Mount Sinai School of Medicine New York, NY 10029-6574 [email protected] Advisors L. Greengard P. Holmes J. Keener J. Keller R. Laubenbacher B.J. Matkowsky A. Mielke C.S. Peskin K.R. Sreenivasan A. Stevens A. Stuart
For further volumes: http://www.springer.com/series/34
Michael E. Taylor
Partial Differential Equations III Nonlinear Equations Second Edition
ABC
Michael E. Taylor Department of Mathematics University of North Carolina Chapel Hill, NC 27599 USA [email protected]
ISSN 0066-5452 ISBN 978-1-4419-7048-0 e-ISBN 978-1-4419-7049-7 DOI 10.1007/978-1-4419-7049-7 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010937758 Mathematics Subject Classification (2010): 35A01, 35A02, 35J05, 35J25, 35K05, 35L05, 35Q30, 35Q35, 35S05 c Springer Science+Business Media, LLC 1996, 2011 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)
To my wife and daughter, Jane Hawkins and Diane Taylor
Contents
Contents of Volumes I and II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .
xi
Preface .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . xiii 13 Function Space and Operator Theory for Nonlinear Analysis . . . . . . . 1 1 Lp -Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 2 2 Sobolev imbedding theorems . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 4 3 Gagliardo–Nirenberg–Moser estimates . . . . . . . . . . . . . .. . . . . . . . . . . . . . 8 4 Trudinger’s inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 14 5 Singular integral operators on Lp . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 17 6 The spaces H s;p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 24 7 Lp -spectral theory of the Laplace operator . . . . . . . . . .. .
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