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Kung-Ching Chang Methods in Nonlinear analysis S ringer Kung-Ching Chang School of mathematical sciences Peking University 10087I Beijing Peoples republic of China Library of Congress Control Number: 200593II37 Mathematics Subject Classification(2000): 47Hoo, 47Jo5, 47Jo7, 47J25, 47J30, 58-oI 58CI5,58E05,49-0I,49JI5,49J35,49J45,49J53,35-0I ISSN I439-7382 ISBN-I0 3-540-24I33-7 Springer Berlin Heidelberg New York ISBN-I3 978-3-540-24I33-I Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law pringer is a part of Springer Science+Business Media c Springer-Verlag Berlin Heidelberg 2005 Printed in The netherlands The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the e of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: by the authors and Tech Books using a Springer ATEX macro package Cover design: design production GmbH, Heidelberg Printed on acid-free paper SPIN: II369295 4I/Tech Books 5432I 0 Preface Nonlinear analysis is a new area that was born and has matured from abun- dant research developed in studying nonlinear problems. In the past thirt years, nonlinear analysis has undergone rapid growth; it has become part of the mainstream research fields in contemporary mathematical analysis Many nonlinear analysis problems have their roots in geometry, astronomy Huid and elastic mechanics, physics, chemistry, biology, control theory, image processing and economics. The theories and methods in nonlinear analysis stem from many areas of mathematics: Ordinary differential equations, partial differential equations, the calculus of variations, dynamical systems, differen- tial geometry, Lie groups, algebraic topology, linear and nonlinear functional analysis, measure theory, harmonic analysis, convex analysis, game theory optimization theory, etc. Amidst solving these problems, many branches are intertwined, thereby advancing each other The author has been offering a course on nonlinear analysis to gradu ate students at Peking University and other universities every two or three years over the past two decades. Facing an enormous amount of material vast numbers of references, diversities of disciplines, and tremendously differ ent backgrounds of students in the audience, the author is always concerned with how much an individual can truly learn, internalize and benefit from a mere semester course in this subjec The authors approach is to emphasize and to demonstrate the most fun- damental principles and methods through important and interesting example from various problems in different branches of mathematics. However, there are technical difficulties: Not only do most interesting problems require back- ground knowledge in other branches of mathematics, but also, in order to solve these problems, many details in argument and in computation should be in- cluded. In this case, we have to get around the real problem, and deal with a simpler one, such that the application of the method is understandable. The author does not always pursue each theory in its broadest generality; instead he stresses the motivation, the success in applications and its limitations P reface The book is the result of many years of revision of the authors lecture notes. Some of the more involved sections were originally used in seminars as introductory parts of some new subjects. However, due to their importance, the materials have been reorganized and supplemented, so that they may be more valuable to the readers In addition. there are notes, remarks, and comments at the end of this book, where important references, recent progress and further reading are presented The author is indebted to Prof. Wang Zhiqiang at Utah State University. Prof. Zhang Kewei at Sussex University and Prof. Zhou Shulin at Peking University for their careful reading and valuable comments on Chaps. 3, 4 and 5 Peking University Kung Ching Chang September, 2003 Contents 1 Linearization 1.1 Differential Calculus in Banach Spaces 1.1. 1 Frechet Derivatives and Gateaux Derivatives 1. 1.2 Nemytscki Operator 1. 1.3 High-Order Derivatives 1.2 Implicit Function Theorem and Continuity Method 12 1.2.1 Inverse Function theorem 12 1.2.2 Applications 17 1.2.3 Continuity method 23 1.3 Lyapunov-Schmidt Reduction and Bifurcation 30 1.3.1 Bifurcation 30 1.3.2 Lyapunov-Schmidt reduction 33 1.3.3 A Perturbation Problem 43 1.3.4 Gluing 47 1.3.5 Transversalit 49 1.4 Hard Implicit Function Theorem 1.4.1 The Small divisor problem 1.4.2 Nash-Moser iteration 2 Fixed-Point theorems 2.1 Order Method 72 2.2 Convex Function and Its Subdifferentials 80 2.2.1 Convex Functions 2.2.2 Subdifferentials 84 2.3 Convexity and compactness 2. 4 Nonexpansive Maps ..104 2.5 Monotone mappings 109 2.6 Maximal Monotone Mapping 120 VIII Contents 3 Degree Theory and Applications 127 3. 1 The Notion of Topological Degree 128 3.2 Fundamental Properties and Calculations of Brouwer Degrees. 137 3.3 Applications of Brouwer Degree ..148 3.3.1 Brouwer Fixed-Point theorem ..148 3.3.2 The Borsuk-Ulam Theorem and Its Consequences 148 3.3.3 Degrees for S. Equivariant Mappings 151 3.3.4 Intersection 153 3.4 Leray-Schauder Degrees 155 3.5 The global bifurcation 164 3.6 Applications ..175 3.6.1 Degree Theory on Closed Convex Sets 175 3.6.2 Positive Solutions and the Scaling Method 180 3.6.3 Krein-Rutman Theory for Positive Linear Operators. 185 3.6.4 Multiple Solutions 189 3.6.5 A Free Boundary Problem 192 3.6.6 Bridging 193 3.7 Extensions 195 3.7.1 Set-Valued mappings 195 3.7.2 Strict Set Contraction Mappings and Condensing Mappings 19 3.7.3 Fredholm Mappings 200 4 Minimization methods 205 4.1 Variational Principles 206 4.1.1 Constraint Problems 206 4.1.2 Euler-Lagrange Equation 209 4.1.3 Dual Variational Principle 212 4.2 Direct method ...216 4.2.1 Fundamental Principle 216 4.2.2 Examples .217 4 The Prescribing gaussian Curvature Problem and the Schwarz symmetric rearrangement 223 4.3 Quasi-Convexity 4.3.1 Weak Continuity and Quasi-Convexity 231 232 4.3.2 Morrey Theorem 237 4.3.3 Nonlinear Elasticity 242 4.4 Relaxation and Young measure 244 4.4.1 Relaxations 245 4.4.2 Young Measure 251 4.5 Other Function Spaces ..260 45.1BⅤ Space 260 4.5.2 Hardy Space and BMo Space 266 4.5.3 Compensation Compactness 271 4.5.4 Applications to the Calculus of variations 274 ontents 4.6 Free Discontinuous Problems 279 4.6.1 T-convergence 279 4.6.2 A Phase Transition problem 280 4.6.3 Segmentation and Mumford-Shah Problem ..284 4.7 Concentration Compactness 289 4.7.1 Concentration Function 289 4.7.2 The Critical Sobolev Exponent and the Best Constants 295 4.8 Minimax Methods ...301 4.8.1 Ekeland Variational Principle 301 4.8.2 Minimax Principle 4.8.3 Applications ..306 5 Topological and Variational Methods 5.1 Morse The 317 5.1.1 Introduction 317 5.1.2 Deformation Theorem 319 5.1.3 Critical Groups 327 5. 1.4 Global Theory 334 5. 1.5 Applications 343 5.2 Minimax Principles(Revisited) 347 5.2. 1 A Minimax Principle 347 5.2.2 Category and Ljusternik-Schnirelmann Multiplicity theorem 349 5.2.3 Cap Product 354 5.2.4 Index theorem 358 5.2.5 Applications 363 5. 3 Periodic Orbits for Hamiltonian System and Weinstein Conjecture 371 5.3.1 Hamiltonian Operator 373 5.3.2 Periodic Solutions 374 5.3.3 Weinstein Conjecture 5.4 Prescribing Gaussian Curvature Problem on s 376 380 5.4.1 The Conformal Group and the best Constant 380 5.4.2 The Palais- Smale sequence 387 5.4.3 Morse Theory for the Prescribing Gaussian Curvature Equation on 5.5 Conley Index Theory 392 5.5.1 Isolated invariant Set ...393 5.5.2 Index Pair and Conley Index 397 5.5. 3 Morse Decomposition on Compact Invariant Sets and Its extension 408 otes 4 References 425 Linearization The first and the easiest step in studying a nonlinear problem is to lineari it. That is, to approximate the initial nonlinear problem by a linear one. Non- linear differential equations and nonlinear integral equations can be seen as nonlinear equations on certain function spaces. In dealing with their lineariza tions, we turn to the differential calculus in infinite-dimensional spaces. The implicit function theorem for finite-dimensional space has been proved very useful in all differential theories: Ordinary differential equations, differentia geometry, differential topology, Lie groups etc. In this chapter we shall see that its infinite-dimensional version will also be useful in partial differential equations and other fields; in particular, in the local existence, in the stability. in the bifurcation, in the perturbation problem, and in the gluing technique etc. This is the contents of sects. 1.2 and 1.3. based on Newton iterations and the smoothing operators, the Nash-Moser iteration, which is motivated by the isometric embeddings of riemannian manifolds into Euclidean spaces and the Kam theory is now a very important tool in analysis. Limited in space and time, we restrict ourselves to introducing only the spirit of the method in Sect. 1.4 1.1 Differential Calculus in Banach Spaces There are two kinds of derivatives in the differential calculus of several vari ables, the gradients and the directional derivatives. We shall extend these two to infinite-dimensional spaces LetX, Y and z be banach spaces, with norms‖·‖x,‖·‖ respectively. If there is no ambiguity, we omit the subscripts. Let u C X be an open set, and let f: U-Y be a map

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