Methods on Nonlinear Elliptic Equations (Aims Series on Differential Equations & Dynamical Systems)

✍ Scribed by Wenxiong Chen, Congming Li

• Publisher: American Institute of Mathematical Sciences
• Year: 2010
• Tongue: English
• Leaves: 311
• Series: Aims Series on Differential Equations & Dynamical Systems 4
• Edition: 1st
• Category: Library

✦ Synopsis

This book serves as a bridge between graduate textbooks and research articles in the area of nonlinear elliptic partial differential equations. Whereas graduate textbooks present basic concepts, the student can hardly get a feel for research by relying solely on such texts; by contrast, whereas journal articles present results on the forefront of research, such texts offer little, if anything, in the way of requisite background material. If this dilemma sounds all too familiar, and you would like to commence hands-on research immediately, this is the book for you; for the purpose of this text is to prepare both graduate students and young mathematicians to readily engage in research and to solve related problems. This volume is self-contained in that it provides both background material and typical methods used in nonlinear analysis, such as: 1) Sobolev Spaces on Euclidean spaces and Riemannian manifolds; 2) Variational methods and critical point theory; 3) Equations on prescribing Gaussian and scalar curvature; 4) Regularity of solutions; 5) Various maximum principles and methods of moving planes. Moreover, it presents new ideas from the research front, including: 1) Regularity lifting by the combined use of contracting and shrinking operators; 2) The method of moving planes in integral forms.

✦ Table of Contents

Cover......Page 1
Title Page......Page 3
Copyright Page......Page 4
Preface......Page 5
Contents......Page 9
1 Introduction to Sobolev Spaces......Page 13
1.1 Distributions......Page 15
1.2 Sobolev Spaces......Page 20
1.3 Approximation by Smooth Functions......Page 22
1.4 Sobolev Embeddings......Page 34
1.5 Compact Embedding......Page 45
1.6.1 Poincare'sInequality......Page 48
1.6.2 The Classical Hardy-Littlewood-Sobolev Inequality......Page 50
2 Existence of Weak Solutions......Page 55
2.1 Second Order Elliptic Operators......Page 56
2.2 Weak Solutions......Page 57
2.3.1 Linear Equations......Page 58
2.3.2 Some Basic Principles in Functional Analysis......Page 59
2.3.3 Existence of Weak Solutions......Page 63
2.4.2 Calculus of Variations......Page 65
2.4.3 Existence of Minimizers......Page 67
2.4.4 Existence of Minimizers Under Constraints......Page 70
2.4.5 Mini-max Critical Points......Page 74
2.4.6 Existence of a Mini-max via the Mountain Pass Theorem......Page 80
3 Regularity of Solutions......Page 89
3.1.1 Newtonian Potentials......Page 91
3.1.2 Uniform Elliptic Equations......Page 98
3.2 W^{2,P} Regularity of Solutions......Page 103
3.2.1 The Case p > 2......Page 105
3.2.2 The Case 1 < p < 2......Page 110
3.2.3 Other Useful Results Concerning the Existence, Uniqueness, and Regularity......Page 112
3.3.1 Bootstrap......Page 113
3.3.2 Regularity Lifting by Contracting Operators......Page 114
3.3.3 Applications to PDEs......Page 115
3.3.4 Applications to Integral Equations......Page 121
3.3.5 Regularity Lifting by Combinations of Contracting and Shrinking Operators......Page 123
3.3.6 Applications to Integral Equations......Page 127
3.3.7 Applications to Fully Nonlinear Systems of Wolff Type.......Page 131
4 Preliminary Analysis on Riemannian Manifolds......Page 135
4.1 Differentiable Manifolds......Page 136
4.2 Tangent Spaces......Page 137
4.3 Riemannian Metrics......Page 140
4.4.1 Curvature of Plane Curves......Page 143
4.4.2 Curvature of Surfaces in R3......Page 144
4.4.3 Curvature on Riemannian Manifolds......Page 145
4.5.1 Higher Order Covariant Derivatives and the Laplace-Beltrami Operator......Page 149
4.5.2 Integrals......Page 151
4.5.3 Equations on Prescribing Gaussian and Scalar Curvaturel......Page 52
4.6 Sobolev Embeddings......Page 153
5.1 Variational Methods in General......Page 157
5.2 Introduction to Prescribing Gaussian Curvature......Page 160
5.3.1 Kazdan and Warner's Results-Method of Lower and Upper Solutions......Page 162
5.3.2 The Limiting Situation......Page 168
5.4.1 Obstructions......Page 174
5.4.2 The Variational Approach and Key Inequalities......Page 175
5.4 3 Existence of Weak Solutions in Subcritical Case......Page 178
5.4.4 A Remedy for Critical Case-Recovering Coerciveness......Page 179
5.4.5 Existence of Weak Solutions in the Critical Case......Page 183
6.1.1 The Yamabe Problem......Page 191
6.1.2 Prescribing Scalar Curvature on S^n......Page 30
6.2 The Variational Approach for the Yamabe Problem......Page 198
6.3.1 Estimate the Values of the Functional......Page 204
6.3.2 The Variational Scheme......Page 214
6.4 The a priori Estimates for Prescribing Scalar Curvature......Page 215
6.4.2 In the Region Where R is Small......Page 216
6.4.3 In the Regions Where R > 0......Page 218
7.1 Introduction......Page 221
7.2 Weak Maximum Principles......Page 226
7.3 The Hopf Lemma and Strong Maximum Principles......Page 230
7.4 Maximum Principles Based on Comparisons......Page 236
7.5 A Maximum Principle for Integral Equations......Page 239
8 Methods of Moving Planes and Moving Spheres......Page 243
8.1 Outline of the Method of Moving Planes......Page 245
8.2.1 Symmetry of Solutions in a Unit Ball......Page 247
8.2.2 Symmetry of Solutions of -Du = un in R"......Page 250
8.2.3 Symmetry of Solutions for -Au = e" in R2......Page 258
8.3.1 The Background......Page 263
8.3.2 The A Priori Estimates......Page 265
8.4.1 The Background......Page 272
8.4.2 Necessary Conditions......Page 274
8.5 Method of Moving Planes in Integral Forms......Page 278
A.1.1 Algebraic and Geometric Notation......Page 287
A.1.2 Notation for Functions and Derivatives......Page 288
A.1.3 Function Spaces......Page 289
A.2 Notation and Basic Facts from Riemannian Geometry......Page 291
A.3 Common Inequalities and Their Proofs......Page 294
A.4 Calderon-Zygmund's Decomposition......Page 296
A.5 The Contracting Mapping Principle......Page 298
A.7 The Proof of Lemma 5.2.1......Page 300
References......Page 303
Index......Page 309