Nonlinearity and Functional Analysis: Lectures on Nonlinear Problems in Mathematical Analysis (Pure and Applied Mathematics, a Series of Monographs and Tex)

Nonlinearity and Functional Analysis is a collection of lectures that aim to present a systematic description of fundamental nonlinear results and their applicability to a variety of concrete problems taken from various fields of mathematical analysis. For decades, great mathematical interest has focused on problems associated with linear operators and the extension of the well-known results of linear algebra to an infinite-dimensional context. This interest has been crowned with deep insights, and the substantial theory that has been developed has had a profound influence throughout the mathematical sciences. This volume comprises six chapters and begins by presenting some background material, such as differential-geometric sources, sources in mathematical physics, and sources from the calculus of variations, before delving into the subject of nonlinear operators. The following chapters then discuss local analysis of a single mapping and parameter dependent perturbation phenomena before going into analysis in the large. The final chapters conclude the collection with a discussion of global theories for general nonlinear operators and critical point theory for gradient mappings. This book will be of interest to practitioners in the fields of mathematics and physics, and to those with interest in conventional linear functional analysis and ordinary and partial differential equations. Nonlinearity and Functional Analysis: Lectures on Nonlinear Problems in Mathematical Analysis......Page 4 Copyright Page......Page 5 Contents......Page 8 Preface......Page 14 Notation and Terminology......Page 18 Suggestions for The Reader......Page 20 PART I: PRELIMINARIES......Page 22 1.1 How Nonlinear Problems Arise......Page 24 1.2 Typical Difficulties Encountered......Page 39 1.3 Facts from Functional Analysis......Page 46 1.4 Inequalities and Estimates......Page 60 1.5 Classical and Generalized Solutions of Differential Systems......Page 68 1.6 Mappings between Finite-Dimensional Spaces......Page 72 2.1 Elementary Calculus......Page 85 2.2 Specific Nonlinear Operators......Page 97 2.3 Analytic Operators......Page 105 2.4 Compact Operators......Page 109 2.5 Gradient Mappings......Page 114 2.6 Nonlinear Fredholm Operators......Page 120 2.7 Proper Mappings......Page 123 Notes......Page 128 PART II: LOCAL ANALYSIS......Page 130 3.1 Successive Approximations......Page 132 3.2 The Steepest Descent Method for Gradient Mappings......Page 148 3.3 Analytic Operators and the Majorant Method......Page 154 3.4 Generalized Inverse Function Theorems......Page 158 Notes......Page 166 4.1 Bifurcation Theory–A Constructive Approach......Page 170 4.2 Transcendental Methods in Bifurcation Theory......Page 184 4.3 Specific Bifurcation Phenomena......Page 194 4.4 Asymptotic Expansions and Singular Perturbations......Page 214 4.5 Some Singular Perturbation Problems of Classical Mathematical Physics......Page 225 Notes......Page 232 PART III: ANALYSIS IN THE LARGE......Page 236 5.1 Linearization......Page 238 5.2 Finite-Dimensional Approximations......Page 252 5.3 Homotopy, the Degree of Mappings, and Its Generalizations......Page 264 5.4 Homotopy and Mapping Properties of Nonlinear Operators......Page 287 5.5 Applications to Nonlinear Boundary Value Problems......Page 304 Notes......Page 317 6.1 Minimization Problems......Page 320 6.2 Specific Minimization Problems from Geometry and Physics......Page 334 6.3 lsoperimetric Problems......Page 345 6.4 Isoperimetric Problems in Geometry and Physics......Page 358 6.5 Critical Point Theory of Marston Morse in Hilbert Space......Page 374 6.6 The Critical Point Theory of Ljusternik and Schnirelmann......Page 387 6.7 Applications of the General Critical Point Theories......Page 396 Notes......Page 409 Appendix A. On Differentiable Manifolds......Page 412 Appendix B. On the Hodge-Kodaira Decomposition for Differential Forms......Page 417 References......Page 420 Index......Page 430 Pure and Applied Mathematics......Page 439