Mathematical Analysis and Numerical Methods for Science and Technology: Volume 2 Functional and Variational Methods

604 Pages·2000·67.85 MB·English

by  Robert Dautray

Mathematical Analysis and Numerical Methods
for Science and Technology
Springer
Berlin
Heidelberg
New York
Barcelona
Hong Kong
London
Milan
Paris
Singapore
Tokyo
Robert Dautray Ja cques-Louis Lions
Mathematical Analysis
and Numerical
Methods for Science
and Technology
Volume 2
Functional and Variational Methods
With the Collaboration of
Michel Artola, Marc Authier, Philippe Benilan,
Michel Cessenat, Jean-Michel Combes, Helene Lanchon,
Bertrand Mercier, Claude Wild, Claude Zuily
Translated from the French by Ian N. Sneddon
, Springer
Robert Dautray
12 rue du Capitaine Scott
75015 Paris, France
Jacques-Louis Lions
College de France
3 rue d'Ulm
75231 Paris Cedex 5, France
Title of the French original edition:
Analyse mathematique et calcul numerique pour les sciences
et les techniques, Masson, S. A.
a
© Commissariat I'Energie Atomique, Paris 1984, 1985
With 20 Figures
Mathematics Subject Classification (1980): 35-XX, 41-XX, 42-XX, 44-XX,
45-XX, 46-XX, 47-XX, 65-XX, 73-XX, 76-XX, 78-XX, 80-XX, 81-XX
Library of Congress Cataloging-in-Publication Data
Dautray, Robert. [Analyse mathematique et caleul numerique pour les sciences et les techniques. English]
Mathematical analysis and numerical methods for science and technology / Raben Dautray. Jacques-Louis Lions; transla
ted from the French by Ian N. Sneddon.
Translation of: Analyse mathematique et caleul numerique pour les sciences et les techniques.
Bibliography: v. 2. p Includes index
Contents: - v. 2 Functional and variational methods!
I. Mathematical analysis. 2. Numerical analysis. I. Lions. Jacques Louis. II. Title.
QA300.D34313 1988 515 - dc 19 88-15089 CIP
ISBN-13: 978-3-540-66098-9 e-ISBN-13: 978-3-642-61566-5
001: 10.1007/978-3-642-61566-5
This work is subject to copyright. All rights are reserved. whether the whole or part of the material is concerned. specifi
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© Springer-Verlag Berlin Heidelberg 1988. 2000
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Typesetting: Macmillan India Limited, Bangalore
SPIN: 10732811 41/3143-5 4 3 2 I -Printed on acid-free paper
Introduction to Volume 2
This second volume (which contains Chaps. III to VII) begins by introducing
some fundamental techniques: first of all, Fourier series (Chap. III) and the
Fourier transform (Appendix on Distributions) and by way of complement, the
Hankel and Mellin transforms (the Laplace transform comes in Chap. XVI,
Vol. 5); finally, for its importance in numerical applications, the method of the
fast Fourier transform. Chapter IV introduces the Sobolev spaces which play
a decisive role in the theory of partial differential equations as well as in
approximation procedures. These chapters III and IV both make use of the
theory of distributions, which is the subject of an Appendix (which can be read
independently).
Chapter V is a study of the linear differential operators in a sufficiently general
context. This setting goes a little beyond the strict needs of the applications
indicated in this work, but the generality introduced allows us better to make
the essentials more clear.We highlight the role of characteristics and the classi
cal classification of linear differential operators into elliptic, parabolic and
hyperbolic operators.
For boundary value problems, a possible formulation is obtained by the use
of unbounded operators (the "domain" of the unbounded operator correspond
ing, for example, to the boundary conditions, supposed homogeneous); this is
why it has seemed reasonable to include (in Chap. VI) a review of the principal
concepts relating to operators in Banach or in Hilbert spaces.
The last chapter of this volume, Chap. VII, introduces the very powerful
variational methods, which, together with the Sobolev spaces, play the most
important role throughout the theory (but which naturally are not the only
ones, and, besides, are not always applied!).
We give below the authors of various contributions, chapter by chapter.
Chapter III: M. Cessenat, B. Mercier, C. Zuily.
Chapter IV: M. Artola, M. Cessenat, C. Zuily.
Chapter V: P. Benilan.
Chapter VI: M. Artola, M. Cessenat, 1.-M. Combes, C. Wild.
Chapter VII: M. Authier, M. Artola, P. Benilan, M. Cessenat, H. Lanchon,
B. Mercier.
Appendix. "Distributions": M. Artola, M. Authier, M. Cessenat.
Finally we mention the partial contributions of the following assistants of the
a
Commissariat I'Energie Atomique: Messrs. Batail, Gambaudo, Giorla, Sznit
man, Verwaerde.
VI Introduction to Volume 2
Practical Guide for the Reader
Designation of subdivisions of the text:
number of a chapter: in Roman numerals
number of part of a chapter: the sign § followed by a numeral
number of section: a numeral following the above
number of a sub-section: a numeral following the above.
Example. II, § 3.5.2 denotes Chapter II, Part 3, section 5, sub-section 2.
The reader wishing to become acquainted rapidly with the mathematical and
numerical essentials of the subject will find them in Vols. 1 and 2 omitting at
a first reading §§ 5, 6, 7 and 8 of Chapter II (vol. 1) and §§ 4, 5 of Chapter V
(vol. 2). These parts are distinguished by an asterisk at the appropriate part of
the text, and also in table of contents.
Table of Contents
Chapter III. Functional Transformations
Introduction . . . . . . . . . . . .
Part A. Some Transformations Useful in Applications 4
§ 1. Fourier Series and Dirichlet's Problem 4
1. Fourier Series . . . . . . . . 4
1.1. Convergence in L2 (1f) 5
1.2. Pointwise Convergence on 1f 5
2. Distributions on 1f and Periodic Distributions 7
2.1. Comparison of [0' (1f) with the Distributions on 1R 7
2.2. Principal Properties of [0' (1f). . . 9
3. Fourier Series of Distributions . . . . 10
4. Fourier Series and Fourier Transforms. 14
5. Convergence in the Sense of Cesaro . . 15
6. Solution of Dirichlet's Problem with the Help of Fourier Series 17
6.1. Dirichlet's Problem in a Disk. . . 17
6.2. Dirichlet's Problem in a Rectangle. 20
§2. The Mellin Transform . . . . . . . 24
1. Generalities . . . . . . . . . . 24
2. Definition of the Mellin Transform 26
3. Properties of the Mellin Transform 28
4. Inverse Mellin Transform. . . . . 30
5. Applications of the Mellin Transform 32
6. Table of Some Mellin Transforms 40
§3. The Hankel Transform. . . . . . 40
1. Generalities . . . . . . . . . 40
2. Introduction to Bessel Functions 42
3. Definition of the Hankel Transform 47
4. The Inversion Formula . . . . . 48
5. Properties of the Hankel Transform 50
6. Application of the Hankel Transform to Partial Differential
Equations . . . . . . . . . . . . . . . . . . . . " 53
VIII Table of Contents
6.1. Dirichlet's Problem for Laplace's Equation in IR~ .
The Case of Axial Symmetry. . . . . . . . . . 53
6.2. Boundary Value Problem for the Biharmonic Equation
in IR~. with Axial Symmetry . 55
7. Table of Some Hankel Transforms 57
Review of Chapter III A . . . . . . . . 57
Part B. Discrete Fourier Transforms and Fast Fourier Transforms 59
§ 1. Introduction . . . . . . . . . . . . . . . . . 59
§2. Acceleration of the Product of a Matrix by a Vector 62
§3. The Fast Fourier Transform of Cooley and Tukey 64
§4. The Fast Fourier Transform of Good-Winograd. 66
§5. Reduction of the Number of Multiplications 69
1. Relation Between the Discrete Fourier Transform
and the Problem of Cyclic Convolution . . . . 69
2. Complexity of the Product of Two Polynomials . 71
3. Application to the Cyclic Convolution of Order 2 72
4. Application to the Cyclic Convolution of Order 3 73
5. Application to the Cyclic Convolution of Order 6 75
§6. Fast Fourier Transform in Two Dimensions. . . 77
§7. Some Applications of the Fast Fourier Transform 78
1. Solution of Boundary Value Problems . . . . 78
2. Regularisation and Smoothing of Functions 81
3. Practical Calculation of the Fourier Transform of a Signal 83
4. Determination of the Spectrum of Certain Finite Difference
Operators and Fast Solvers for the Laplacian 84
Review of Chapter III B . . . . . . . . . . . . . . . . . .. 91
Chapter IV. Sobolev Spaces
Introduction . . . . . . 92
§1. Spaces HI (Q), /!"'(Q) 92
§2. The Space H'(IR") .. 96
1. Definition and First Properties 96
2. The Topological Dual of HS (IR") 98
3. The Equation (-Ll +P)u=fin IR", kEIR\{O} 100
§3. Sobolev's Embedding Theorem . . . . . . . . 100
§4. Density and Trace Theorems for the Spaces Hm(Q),
(mEN*=N\{O}) .............. . 102
Table of Contents IX
1. A Density Theorem. . . . . . . . . . 102
2. A Trace Theorem for HI OR"+.) . . . . . 107
3. Traces of the Spaces Hrn(IR"+.) and Hrn(Q) 113
4. Properties of m-Extension . . 114
§5. The Spaces H-m(Q) for all mEN . 120
§6. Compactness . . . . . . . . . . 123
§7. Some Inequalities in Sobolev Spaces 125
1. Poincare's Inequality for H6(Q) (resp. H[;'(Q». 125
2. Poincare's Inequality for HI (Q) . 127
3. Convexity Inequalities for Hm (Q) 133
§8. Supplementary Remarks . . 138
1. Sobolev Spaces wm,p(Q) . 138
1.1. Definitions .. . . 138
1.2. So bolev Injections 139
1.3. Trace Theorems for the Spaces Wm,P(Q) 140
2. Sobolev Spaces with Weights 141
2.1. Unbounded Open Sets 141
2.2. Polygonal Open Sets 141
Review of Chapter IV . . , , , . 142
Appendix: The Spaces HS(r) with r the "Regular" Boundary
of an Open Set Q in IRn . , . , , . , , , . . . . . . . . 143
Chapter V. Linear Differential Operators
Introduction . . . . . . . . . . . . . 148
§ 1. Generalities on Linear Differential Operators . . . 149
1. Characterisation of Linear Differential Operators 149
2. Various Definitions . . . . . . . . . . . . . 152
2.1. Leibniz's Formula .......... . 152
2.2. Transpose of a Linear Differential Operator 153
2.3. Order of a Linear Differential Operator 154
3. Linear Differential Operator on a Manifold 155
4. Characteristics . . . . . . . 157
4.1. Concept of Characteristics . . . 157
4.2. Bicharacteristics . . . . . . . 159
5. Operators with Analytic Coefficients.
Theorems of Cauchy-Kowalewsky and of Holmgren 163
§2. Linear Differential Operators with Constant Coefficients 170
1. Study of a I.d.o. with Constant Coefficients
by the Fourier Transform . . . . . . . . . . . . 171