Variational Methods in Mathematics, Science and Engineering
The impulse which led to the writing of the present book has emerged from my many years of lecturing in special courses for selected students at the College of Civil Engineering of the Tech nical University in Prague, from experience gained as supervisor
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Variational Methods in Mathematics, Science and Engineering
Variational Methods in Mathematics, Science and Engineering Prof. RNDr Karel Rektorys, DrSc
D. REIDEL PUBLISHING COMPANY DORDRECHT-HOLLAND I BOSTON-U.S.A.
Library of Congress Cataloging in Publication Data
Rektorys, Karel. Variational methods in mathematics, science, and engineering. Bibliography: Includes index. 1. Calculus of variations. 2. Hilbert space. 3. Differential equations - Numerical solutions. 4. Boundary value problems - Numerical solutions. I. Title. 74-80530 QA315. R44 515'7 ISBN-13: 978-94-011-6452-8 e-ISBN-13: 978-94-011-6450-4 DOl: 10.1007/978-94-011-6450-4
Translated from the Czech by Michael Basch, 1975 Published by D. Reidel Publishing Company, P. O. Box 17, Dordrecht, Holland in co-edition with SNTL - Publishers of Technical Literature - Prague Sold and distributed in the USA, Canada and Mexico by D. Reidel Publishing Company, Inc., Lincoln Building, 160 Old Derby Street, Hingham, Mass. 02043, USA All Rights Reserved Copyright © 1977 by Karel Rektorys Softcover reprint of the hardcover 1st edition 1977 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any informational storage and retrieval system, without written permission from the copyright owner
To my wife
CONTENTS
Preface 11 Notation Frequently Used 15 Chapter 1. Introduction 17 Part 1. HILBERT SPACE Chapter 2. Inner Product of Functions. Norm, Metric 21 Chapter 3. The Space L2 32 Chapter 4.
Convergence in the Space L 2 (G) (Convergence in the Mean). Complete Space. Separable Space 38 a) Convergence in the space L 2 (G) 38 b) Completeness 41 c) Density. Separability 44
Chapter 5. Orthogonal Systems in L 2 (G) 47 a) Linear dependence and independence in LiG) 47 b) Orthogonal and orthonormal systems in L 2 (G) 51 c) Fourier series. Complete systems. The Schmidt orthonormalization 54 d) Decomposition of L 2 (G) into orthogonal subspaces 62 e) Some properties of the inner product 64 Chapter 6. Hilbert Space 66 a) Pre-Hilbert space. Hilbert space 66 b) Linear dependence and independence in a Hilbert space. Orthogonal systems, Fourier series 74 c) Orthogonal subspaces. Some properties of the inner product 78 d) The complex Hilbert space 79 Chapter 7. Some Remarks to the Preceding Chapters. Normed Space, Banach Space 81 Chapter 8. Operators and Functionals, especially in Hilbert Spaces 86 a) Operators in Hilbert spaces 87 b) Symmetric, positive, and positive definite operators. Theorems on density 98 c) Functionals. The Riesz theorem 109 Part II. VARIATIONAL METHODS Chapter 9.
Theorem on the Minimum of a Quadratic Functional and its Consequences 113
Chapter 10. The Space HA 121 Chapter II. Existence of the Minimum of the Functional Fin the Space H A . Generalized Solutions 133
7
VARIATIONAL METHODS IN MATHEMATICS, SCIENCE AND ENGINEERING Chapter 12. The Method of Orthonormal Series. Example 146 Chapter 13. :rhe Ritz Method 153 Chapter 1
