Functional Calculus of Pseudodifferential Boundary Problems
Pseudodifferential methods are central to the study of partial differential equations, because they permit an "algebraization." A replacement of compositions of operators in n-space by simpler product rules for thier symbols. The main purpose of this book
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Series Editors Hyman Bass Joseph Oesterle Alan Weinstein
GerdGrubb
Functional Calculus of Pseudodifferential Boundary Problems Second Edition
Springer Science+Business Media, LLC
Gerd Grubb K~benhavns Universitets Matematiske Institut 2100 K~benhavn 0 Oenmark
Library of Congress Cataloging-in-Publication Data Grubb, Gerd Functional calculus of pseudodifferential boundary problems / Gerd Grubb. --[2nd ed.] p. cm. (Progress in Mathematics ; v. 65) lncludes bibliographical references and index. ISBN 978-1-4612-6895-6 ISBN 978-1-4612-0769-6 (eBook) DOI 10.1007/978-1-4612-0769-6 1. Pseudodifferential operators. 2. Boundary value problems. I. Title. II. Series : Progress in mathematics (Boston, Mass.) VoI. 65. QA 329.7.078 1996 95-40159 515'.7242--dc20 CIP
Printed on acid-free paper © 1996 Springer Science+Business Media New York Originally published by Birkhäuser Boston in 1996 Softcover reprint ofthe hardcover 2nd edition 1996 Copyright is not claimed for works of V.S. Government employees. AlI rights reserved. No part ofthis publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permis sion of the copyright owner. Permis sion to photocopy for internal or personal use of specific dients is granted by Springer Science+Business Media, LLC. for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of$6.00 percopy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, V.S.A. Special requests should be addressed directly to Springer Science+Business Media, LLC.
ISBN 978-1-4612-6895-6 Typeset by the author in TeX.
9876543210
CONTENTS
Introduction Chapter 1. Standard pseudodifferential boundary problems and their realizations 1.1. Introductory remarks 1.2. The calculus of pseudo differential boundary problems 1.3. Green's formula 1.4. Realizations and normal boundary conditions 1.5. Parameter-ellipticity and parabolicity 1.6. Adjoints 1. 7. Semiboundedness and coerciveness Chapter 2. The calculus of parameter-dependent operators 2.1. Parameter-dependent pseudo differential operators 2.2. The transmission condition 2.3. Parameter-dependent boundary symbols 2.4. Operators and kernels 2.5. Continuity, auxiliary operators 2.6. Composition of boundary symbol operators 2.7. Compositions in general 2.8. Extension of mapping properties, negative class 2.9. Strictly homogeneous symbols Chapter 3. Parametrix and resolvent constructions 3.1. Ellipticity. Auxiliary reductions 3.2. The parametrix construction 3.3. The resolvent of a realization 3.4. Other special cases Chapter 4. Some applications 4.1. Evolution problems 4.2. The heat operator
1 15 15
19 36 44 54 74 98 126
126 158 187 204 231 244 264 277 284 292 292 307 336 358 368 368 373
VI
CONTENTS
4.3. 4.4. 4.5. 4.6. 4.7.
An index formula Complex powers Spectral asymptotics Implicit eigenvalue problems Singular perturbations
Appendix. Various prerequisites
