Fattorini H.O., Kerber A. The Cauchy Problem
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The Cauchy Problem
GIAN-CARLO ROTA, Editor
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
Volume Section
I LUIS A. SANTALO
Integral Geometry and Geometric Probability,
1976 (2nd printing, with revisions, 1979)
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with Applications to Decisionmaking,
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Angular Momentum in Quantum Physics:
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GIAN-CARLO ROTA, Editor
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
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GIAN-CARLO ROTA, Editor
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
Volume
Section
18
H. O. FATTORINI
Analysis
The Cauchy Problem,
1983
19
G. G. LORENTZ, K. JETTER, and
Interpolation
S. D. RIEMENSCHNEIDER
and
Birkhoff Interpolation, 1983
Approximation
20
RUDOLF LIDL and
HARALD NIEDERREITER
Analysis
Finite Fields, 1983
Other volumes in preparation
ENCYCLOPEDIA
OF MATHEMATICS
and Its Applications
GIAN-CARLO ROTA, Editor
Department of Mathematics
Massachusetts Institute of Technology
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ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS, 18.
The Cauchy Problem
H. O. Fattorini
University Of California, Los Angeles
CAMBRIDGE
UNIVERSITY PRESS
PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
The Pitt Building, Trumpington Street, Cambridge, UK
CAMBRIDGE UNIVERSITY PRESS
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m 1983 Cambridge University Press
This book is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 1983
A catalogue record for this book is available from the British Library
Library of Congress Cataloging in Publication Data
Fattorini, H. O. (Hector 0.), 1938-
The Cauchy problem
(Encyclopaedia of Mathematics and its Applications; v. 18)
Bibliography; p
Includes index
1. Cauchy problem. 2. Initial value problems.
1. Title, II. Series.
QA378.F37 1983
515.3'54 82-22669
ISBN 0 521 30238 2 hardback
Transferred to digital printing 1999
Contents
Editor's Statement
......................................xiii
Foreword
.............................................xv
Preface ..............................................xix
Chapter 0
Elements of Functional Analysis
......................1
1
Banach and Hilbert Spaces. Bounded Operators.
Integration of Vector-Valued Functions ................1
2 Linear Functionals: The Dual Space. Vector-Valued
Analytic Functions ...............................
4
3 Unbounded Operators; the Resolvent. Closed Operators ....6
4 Adjoints
.....................................
11
5 Generalized Sequences. Weak Convergence
.............
15
6 Normal, Symmetric, Self-Adjoint, and Unitary
Operators in Hilbert Space
........................
16
7
Some Special Spaces and Their Duals .................20
8
Convolution and Mollifiers. Sobolev Spaces ............23
Chapter 1
The Cauchy Problem for Some Equations of
Mathematical Physics: The Abstract Cauchy Problem ...... 26
1.1 The Heat Equation in a Square .....................27
1.2
The Abstract Cauchy Problem ...................... 29
ix
x
Contents
1.3 The Diffusion Equation in a Square .................. 32
1.4 The Schrodinger Equation
......................... 39
1.5 The Cauchy Problem in (- oo, oo) ...................43
1.6
The Maxwell Equations ..........................43
1.7 Miscellaneous Notes ............................. 54
Chapter 2
Properly Posed Cauchy Problems: General Theory ........62
2.1 The Cauchy Problem in t > 0 ......................
63
2.2
The Cauchy Problem in - oo < t < oo. The Adjoint
Equation .....................................71
2.3
Semigroup Theory
..............................80
2.4 The Inhomogeneous Equation ......................
86
2.5
Miscellaneous Comments .........................91
Chapter 3
Dissipative Operators and Applications ...............117
3.1
Dissipative Operators
........................... 117
3.2 Ordinary Differential Operators in the Whole Line
....... 125
3.3 Ordinary Differential Operators in a Closed Interval.
Semi-Infinite Intervals
........................... 135
3.4
Ordinary Differential Operators in a Compact Interval ....142
3.5 Symmetric Hyperbolic Systems in the Whole Space
...... 146
3.6 Isometric Propagators and Conservative Operators.
Dissipative Operators in Hilbert Space
............... 154
3.7 Differential Equations in Banach Lattices. Positive
Solutions. Dispersive Operators
.................... 159
3.8
Miscellaneous Comments
........................ 166
Chapter 4
Abstract Parabolic Equations: Applications to
Second Order Parabolic Equations ..................172
4.1 Abstract Parabolic Equations
......................173
4.2 Abstract Parabolic Equations; Analytic Propagators ......179
4.3 Applications to Ordinary Differential Operators ........ 187
4.4 Second Order Partial Differential Operators.
Dissipativity
..................................199
4.5
Second Order Partial Differential Operators.
Assignation of Boundary Conditions.................204
4.6 Second Order Partial Differential Operators.
Construction of m-Dissipative Extensions in L2 .........214
4.7 Regularity Theorems
............................
222
4.8 Construction of m-Dissipative Extensions in LP(Q) and
C(Q) .......................................235
4.9 Analyticity of Solution Operators ...................255