Precise Spectral Asymptotics for Elliptic Operators Acting in Fiberings over Manifolds with Boundary

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Victor Ivrii

Precise Spectral Asymptotics for Elliptic Operators Acting in Fiberings over Manifolds with Boundary

Spri nger-Verlag Berlin Heidelberg New York Tokyo 1984

Author

Victor Ivrii Dept of Mathematics, Institute of Mining and Metallurgy Magnitogorsk 455000, USSR

Consulting Editor

Olga A. Ladyzhenskaya Leningrad Branch of the VA Steklov Mathematical Institute Fontanka 27, Leningrad 191011, USSR

AMS Subject Classification (1980): 35P20, 58G17, 58G20, 58G25 ISBN 3-540-13361-5 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13361-5 Springer-Verlag New York Heidelberg Berlin Tokyo

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.

© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

TABLE OF CONTENTS

Introduction . • . . . . . . . . . . . . . . . . . . . . • • IV

PART I. THE ASYMPTOTICS FOR SECOND-ORDER OPERATORS O. Main theorems

. . . . . . . . . . . • .

1. The L.Boutet de Monvel operator algebra and selfadjoint projectors. • . . .

. . . • . . 19

2. The "Wave" equation: finite speed of propagation of singularities . . . .

44

3. The normality of the "great" singularity. .

61

4. Calculation of the "great" singularity 5. Proofs of the main theorems . . . . •

....

Appendices A - E . . . . . . . . . . . . . . . . . Part IL THE ASYMPTOTICS FOR MISCELLANEOUS PROBLEMS

96 150 160 177

6. The asymptotics for first-order operators

177

7. The asymptotics for second-order spectral problems. . . . . • . . . . . . . • . . . . . . .

183

8. The asymptotics for higher order spectral problems related to the Laplace-Beltrami operator. . . . .

190

Appendix F . . . . . . . . . . . . . . . . . . . .

225

List of notations

226

Subject index

226

References . .

229

INTRODUCTION This book is devoted to the determination of the precise asymptotics for eigenvalues of certain elliptic selfadjoint operators acting in fiberings over compact manifolds with boundary and for more general elliptic selfadjoint spectral problems. The preoise asymptotics for restriction to the diagonal of the Schwartz kernels of the corresponding spectral projectors is derived too. These asymptotics for closed manifolds were determined in author's paper [57] using the same method; therefore onecan consider [57J as a simple and short introduction to the methods and ideas of this book. More general results were obtained, for example, in ()2 ­ 35], but with much weaker remainder estimates. In Part I we derive these asymptotics for the second­order elliptic selfadjoint differential operators acting in fiberings over manifolds with boundary on which an ell