Topics in functional analysis and applications

Cover......Page 1
Title page......Page 3
Copyright page......Page 4
Preface......Page 5
Notations......Page 9
Contents......Page 11
1.1 Introduction......Page 13
1.2 Test Functions and Distributions......Page 16
1.3 Some Operations with Distributions......Page 21
1.4 Supports and Singular Supports of Distributions......Page 24
1.5 Convolution of Functions......Page 29
1.6 Convolution of Distributions......Page 34
1.7 Fundamental Solutions......Page 41
1.8 The Fourier Transform......Page 45
1.9 The Schwartz Space, $\\mathcal{S}$......Page 48
1.10 The Fourier Inversion Formula......Page 53
1.11 Tempered Distributions......Page 55
Comments......Page 57
Exercises 1......Page 60
2.1 Definition and Basic Properties......Page 63
2.2 Approximation by Smooth Functions......Page 69
2.3 Extension Theorems......Page 74
2.4 Imbedding Theorems......Page 84
2.5 Compactness Theorems......Page 93
2.6 Dual Spaces, Fractional Order Spaces and Trace Spaces......Page 100
2.7 Trace Theory......Page 107
Comments......Page 115
Exercises 2......Page 118
3.1 Some Abstract Variational Problems......Page 123
3.2 Examples of Elliptic Boundary Value Problems......Page 130
3.3 Regularity of Weak Solutions......Page 149
3.4 An Example of the Galerkin Method......Page 153
3.5 Maximum Principles......Page 155
3.6 Eigenvalue Problems......Page 159
3.7 Introduction to the Finite Element Method......Page 164
Comments......Page 172
Exercises 3......Page 174
4.1 Unbounded Linear Operators......Page 178
4.2 The Exponential Map......Page 182
4.3 $C_0$-Semigroups......Page 184
4.4 The Hille-Yosida Theorem......Page 190
4.5 Contraction Semigroups on Hilbert Spaces......Page 200
4.6 The Heat Equation......Page 207
4.7 The Wave Equation......Page 211
4.8 The Schrodinger Equation......Page 215
4.9 The Inhomogeneous Equation......Page 217
Comments......Page 220
Exercises 4......Page 222
5.1 Introduction......Page 226
5.2 Some Fixed Point Theorems......Page 227
5.3 The Galerkin Method......Page 236
5.4 Monotone Iterations......Page 239
5.5 Variational Methods......Page 243
5.6 Pohozaev\'s Identity......Page 249
Comments......Page 252
Exercises 5......Page 255
APPENDIX 1 Partition of Unity......Page 257
APPENDIX 2 The Topology on $\\mathcal{D}(\\Omega)$......Page 260
APPENDIX 3 Calculus in Banach Spaces......Page 262
APPENDIX 4 Stampacchia\'s Theorem......Page 268
References......Page 273
Index......Page 277