Methods of Applied Mathematics
Todd Arbogast and Jerry L. Bona
Department of Mathematics, and
Institute for Computational Engineering and Sciences
The University of Texas at Austin
Copyright 1999–2001, 2004–2005, 2007–2008 (corrected version)
by T. Arbogast and J. Bona.
,
, Contents
Chapter 1. Preliminaries 5
1.1. Elementary Topology 5
1.2. Lebesgue Measure and Integration 13
1.3. Exercises 23
Chapter 2. Normed Linear Spaces and Banach Spaces 27
2.1. Basic Concepts and Definitions. 27
2.2. Some Important Examples 34
2.3. Hahn-Banach Theorems 43
2.4. Applications of Hahn-Banach 48
2.5. The Embedding of X into its Double Dual X ∗∗ 52
2.6. The Open Mapping Theorem 53
2.7. Uniform Boundedness Principle 57
2.8. Compactness and Weak Convergence in a NLS 58
2.9. The Dual of an Operator 63
2.10. Exercises 66
Chapter 3. Hilbert Spaces 73
3.1. Basic Properties of Inner-Products 73
3.2. Best Approximation and Orthogonal Projections 75
3.3. The Dual Space 78
3.4. Orthonormal Subsets 79
3.5. Weak Convergence in a Hilbert Space 86
3.6. Exercises 87
Chapter 4. Spectral Theory and Compact Operators 89
4.1. Definitions of the Resolvent and Spectrum 90
4.2. Basic Spectral Theory in Banach Spaces 91
4.3. Compact Operators on a Banach Space 93
4.4. Bounded Self-Adjoint Linear Operators on a Hilbert Space 99
4.5. Compact Self-Adjoint Operators on a Hilbert Space 104
4.6. The Ascoli-Arzelà Theorem 107
4.7. Sturm Liouville Theory 109
4.8. Exercises 122
Chapter 5. Distributions 125
5.1. The Notion of Generalized Functions 125
5.2. Test Functions 127
5.3. Distributions 129
5.4. Operations with Distributions 133
3
, 4 CONTENTS
5.5. Convergence of Distributions and Approximations to the Identity 138
5.6. Some Applications to Linear Differential Equations 140
5.7. Local Structure of D0 148
5.8. Exercises 148
Chapter 6. The Fourier Transform 151
6.1. The L1 (Rd ) Theory 153
6.2. The Schwartz Space Theory 157
6.3. The L2 (Rd ) Theory 162
6.4. The S 0 Theory 164
6.5. Some Applications 170
6.6. Exercises 172
Chapter 7. Sobolev Spaces 177
7.1. Definitions and Basic Properties 177
7.2. Extensions from Ω to Rd 181
7.3. The Sobolev Imbedding Theorem 185
7.4. Compactness 191
7.5. The H s Sobolev Spaces 193
7.6. A Trace Theorem 198
7.7. The W s,p (Ω) Sobolev Spaces 203
7.8. Exercises 204
Chapter 8. Boundary Value Problems 207
8.1. Second Order Elliptic Partial Differential Equations 207
8.2. A Variational Problem and Minimization of Energy 210
8.3. The Closed Range Theorem and Operators Bounded Below 213
8.4. The Lax-Milgram Theorem 215
8.5. Application to Second Order Elliptic Equations 219
8.6. Galerkin Approximations 224
8.7. Green’s Functions 226
8.8. Exercises 229
Chapter 9. Differential Calculus in Banach Spaces and the Calculus of Variations 233
9.1. Differentiation 233
9.2. Fixed Points and Contractive Maps 241
9.3. Nonlinear Equations 245
9.4. Higher Derivatives 252
9.5. Extrema 256
9.6. The Euler-Lagrange Equations 259
9.7. Constrained Extrema and Lagrange Multipliers 265
9.8. Lower Semi-Continuity and Existence of Minima 269
9.9. Exercises 273
Bibliography 279
Todd Arbogast and Jerry L. Bona
Department of Mathematics, and
Institute for Computational Engineering and Sciences
The University of Texas at Austin
Copyright 1999–2001, 2004–2005, 2007–2008 (corrected version)
by T. Arbogast and J. Bona.
,
, Contents
Chapter 1. Preliminaries 5
1.1. Elementary Topology 5
1.2. Lebesgue Measure and Integration 13
1.3. Exercises 23
Chapter 2. Normed Linear Spaces and Banach Spaces 27
2.1. Basic Concepts and Definitions. 27
2.2. Some Important Examples 34
2.3. Hahn-Banach Theorems 43
2.4. Applications of Hahn-Banach 48
2.5. The Embedding of X into its Double Dual X ∗∗ 52
2.6. The Open Mapping Theorem 53
2.7. Uniform Boundedness Principle 57
2.8. Compactness and Weak Convergence in a NLS 58
2.9. The Dual of an Operator 63
2.10. Exercises 66
Chapter 3. Hilbert Spaces 73
3.1. Basic Properties of Inner-Products 73
3.2. Best Approximation and Orthogonal Projections 75
3.3. The Dual Space 78
3.4. Orthonormal Subsets 79
3.5. Weak Convergence in a Hilbert Space 86
3.6. Exercises 87
Chapter 4. Spectral Theory and Compact Operators 89
4.1. Definitions of the Resolvent and Spectrum 90
4.2. Basic Spectral Theory in Banach Spaces 91
4.3. Compact Operators on a Banach Space 93
4.4. Bounded Self-Adjoint Linear Operators on a Hilbert Space 99
4.5. Compact Self-Adjoint Operators on a Hilbert Space 104
4.6. The Ascoli-Arzelà Theorem 107
4.7. Sturm Liouville Theory 109
4.8. Exercises 122
Chapter 5. Distributions 125
5.1. The Notion of Generalized Functions 125
5.2. Test Functions 127
5.3. Distributions 129
5.4. Operations with Distributions 133
3
, 4 CONTENTS
5.5. Convergence of Distributions and Approximations to the Identity 138
5.6. Some Applications to Linear Differential Equations 140
5.7. Local Structure of D0 148
5.8. Exercises 148
Chapter 6. The Fourier Transform 151
6.1. The L1 (Rd ) Theory 153
6.2. The Schwartz Space Theory 157
6.3. The L2 (Rd ) Theory 162
6.4. The S 0 Theory 164
6.5. Some Applications 170
6.6. Exercises 172
Chapter 7. Sobolev Spaces 177
7.1. Definitions and Basic Properties 177
7.2. Extensions from Ω to Rd 181
7.3. The Sobolev Imbedding Theorem 185
7.4. Compactness 191
7.5. The H s Sobolev Spaces 193
7.6. A Trace Theorem 198
7.7. The W s,p (Ω) Sobolev Spaces 203
7.8. Exercises 204
Chapter 8. Boundary Value Problems 207
8.1. Second Order Elliptic Partial Differential Equations 207
8.2. A Variational Problem and Minimization of Energy 210
8.3. The Closed Range Theorem and Operators Bounded Below 213
8.4. The Lax-Milgram Theorem 215
8.5. Application to Second Order Elliptic Equations 219
8.6. Galerkin Approximations 224
8.7. Green’s Functions 226
8.8. Exercises 229
Chapter 9. Differential Calculus in Banach Spaces and the Calculus of Variations 233
9.1. Differentiation 233
9.2. Fixed Points and Contractive Maps 241
9.3. Nonlinear Equations 245
9.4. Higher Derivatives 252
9.5. Extrema 256
9.6. The Euler-Lagrange Equations 259
9.7. Constrained Extrema and Lagrange Multipliers 265
9.8. Lower Semi-Continuity and Existence of Minima 269
9.9. Exercises 273
Bibliography 279
