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
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© 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
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