439 Pages·1977·19.28 MB·English

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NONLINEARITY AND FUNCTIONAL ANALYSIS Lectures on Nonlinear Problems in Mathematical Analysis This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: SAMUEL EILENBERANCD HYMANBA SS A list of recent titles in this series appears at the end of this volume. N0 NL IN EARITY AND FUNCTIONAL ANALYSIS Lectures on Nonlinear Problems in Mathematical Analysis Melvin S. Berger Belfer Graduate School Yeshiva University New York, New York W ACADEMIC PRESS New York San Francisco London 1977 A Subsidiary of Harcourt Brace Jovanovich, Publishers COPYRIGH0T 1 977, BY ACADEMIPCR ESS,I NC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003 United KitiPdom Edition oublished bv ACADEM~CP RESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1 Library of Congress Cataloging in Publication Data Berger, Melvyn, Date Nonlinearity and functional analysis. (Pure and applied mathematics) Bibliography: p. Includes index. 1. Mathematical analysis. 2. Nonlinear theories. I. Title. QA300.B45 8 515'.7 76 -26 0 39 ISBN 0-12-090350-4 PRINTED IN THE UNITED STATES OF AMERICA Section 6.2A, pp. 313-317, is reprinted with permission of the publisher, American Mathematical Society, from the Pro- ceedings of SYMPOSIA IN PURE MATHEMATICS, Copyright @ 1970, Volume XVIII, Part I, pp. 22-24. To the memory of my father, Abraham Berger How manifold are Thy works, 0 Lord! In wisdom hast Thou made them all. PSALMS 104 The approach to a more profound knowledge of the basic principles of physics is tied up with the most intricate mathematical methods. A. EINSTEIN All depends, then, on finding out these easier problems, and on solving them by means of devices as perfect as possible and of concepts capable of generalization. D. HILBERT CONTENTS Preface xiii Noiation and Terminology xvii Suggestions for ihe Reader xix PART I PRELIMINARIES Chapter 1 Background Material I. 1 How Nonlinear Problems Arise 3 1. IA Differential-geometric sources 3 l.lB Sources in mathematical physics 9 I Classical mathematical physics 10 I1 Contemporary mathematical physics 14 I.1C Sources from the calculus of variations 16 1.2 Typical Difficulties Encountered 18 1.2A Inherent difficulties 18 1.28 Nonintrinsic difficulties 21 1.3 Facts from Functional Analysis 25 1.3A Banach and Hilbert spaces 25 I .3B Some useful Banach spaces 26 1.3C Bounded linear functionals and weak convergence 30 1.3D Compactness 31 1.3E Bounded linear operators 32 1.3F Special classes of bounded linear operators 35 1.4 Inequalities and Estimates 39 1.4A The spaces W,,(Q) (1 I p 5 a) 40 1.4B The spaces W,",,(RN) and W,,,,,(Q) (m 2 1, an integer, and Isp<=) 44 I .4C Estimates for linear elliptic differential operators 45 1.5 Classical and Generalized Solutions of Differential Systems 41 1.5A Weak solutions in Wm,, 48 1.5B Regularity of weak solutions for semilinear elliptic systems 49 1.6 Mappings between Finite-Dimensional Spaces 51 1.6A Mappings between Euclidean spaces 52 1.6B Homotopy invariants 54 1.6C Homology and cohomology invariants 56 vii ... Vlll CONTENTS Notes 59 Chapter 2 Nonlinear Operators 2. I Elementary Calculus 64 2.IA Boundedness and continuity 64 2. IB Integration 65 2.IC Differentiation 61 2.1D Multilinear operators 70 2,lE Higher derivatives 72 2.2 Specific Nonlinear Operators 76 2.2A Composition operators 76 2.2B Differential operators 71 2.2C Integral operators 79 2.2D Representations of differential operators 80 2.3 Analytic Operators 84 2.3A Equivalent definitions 84 2.3B Basic properties 88 2.4 Compact Operators 88 2.4A Equivalent definitions 89 2.48 Basic properties 90 2.4C Compact differential operators 92 2.5 Gradient Mappings 93 2.5A Equivalent definitions 94 2.5B Basic properties 95 2.5C Specific gradient mappings 97 2.6 Nonlinear Fredholrn Operators 99 2.6A Equivalent definitions 99 2.68 Basic properties 100 2.6C Differential Fredholm operators 101 2.7 Proper Mappings 102 2.7A Equivalent definitions 102 2.78 Basic properties 103 2.7C Differential operators as proper mappings 105 Notes 107 PART II LOCAL ANALYSIS Chapter 3 Local Analysis of a Single Mapping 3. I Successive Approximations 111 3.1A The contraction mapping principle 111 3.IB The inverse and implicit function theorems 1 I3 3.1C Newton’s method 1 I6 3.1D A criterion for local surjectivity 118 3.1E Application to ordinary differential equations 1 I9 3. IF Application to isoperirnetric problems 122 CONTENTS ix 3. IG Application to singularities of mappings 125 3.2 The Steepest Descent Method for Gradient Mappings 127 3.2A Continuous descent for local minima I 28 3.2B Steepest descent for isoperimetric variational problems 129 3.2C Results for general critical points 130 3.2D Steepest descent for general smooth mappings 132 3.3 Analytic Operators and the Majorant Method 133 3.3A Heuristics 133 3.3B An analytic implicit function theorem 134 3.3C Local behavior of complex analytic Fredholm operators 136 3.4 Generalized Inverse Function Theorems 137 3.4A Heuristics 137 3.4B A result of J. Moser 138 3.4C Smoothing operators 141 3.4D Inverse function theorems for local conjugacy problems 142 Notf !S 145 Chapter 4 Parameter Dependent Perturbation Phenomena 4.1 Bifurcation Theory-A Constructive Approach 149 4. IA Definitions and basic problems 150 4. IB Reduction to a finite-dimensional problem 154 4.1C The case of simple multiplicity 155 4.1D A convergent iteration scheme 158 4.IE The case of higher multiplicity 161 4.2 Transcendental Methods in Bifurcation Theory 163 4.2A Heuristics 163 4.2B Brouwer degree in bifurcation theory 164 4.2C Elementary critical point theory 167 4.2D Morse type numbers in bifurcation theory 171 4.3 Specific Bifurcation Phenomena 173 4.3A Periodic motions near equilibrium points in the restricted three- body problem 173 4.38 Buckling phenomena in nonlinear elasticity 177 4.3C Secondary steady flows for the Navier-Stokes equation 183 4.3D Bifurcation of complex structures on compact complex manifold< I 88 4.4 Asymptotic Expansions and Singular Perturbations 193 4.4A Heuristics 193 4.48 The validity of formal symptotic expansions 194 m,) 4.4C Application to the semilinear Dirichlet problem 200 4.5 Some Singular Perturbation Problems of Classical Mathematical Physics 204 4.SA Perturbation of an anharmonic oscillator by transient forces 205 4.58 The membrane approximation in nonlinear elasticity 206 4.5C Perturbed Jeffrey-Hamel flows of a viscous fluid 207 Notes 21 1

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