Continuous Functions

This book is the second of a set dedicated to the mathematical tools used in partial differential equations derived from physics. It presents the properties of continuous functions, which are useful for solving partial differential equations, and, more particularly, for constructing distributions valued in a Neumann space. The author examines partial derivatives, the construction of primitives, integration and the weighting of value functions in a Neumann space. Many of them are new generalizations of classical properties for values in a Banach space. Simple methods, semi-norms, sequential properties and others are discussed, making these tools accessible to the greatest number of students – doctoral students, postgraduate students – engineers and researchers, without restricting or generalizing the results.

Jacques Simon is Honorary Research Director at CNRS. His research focuses on Navier-Stokes equations, particularly in shape optimization and in the functional spaces they use.

Introduction ix

Familiarization with Semi-normed Spaces xiii

Notations xv

Chapter 1. Spaces of Continuous Functions 1

1.1. Notions of continuity 1

1.2. Spaces C(Ω;E), C_b(Ω;E), C_K(Ω;E), C(Ω;E) and C_b(Ω;E) 3

1.3. Comparison of spaces of continuous functions 6

1.4. Sequential completeness of spaces of continuous functions 10

1.5. Metrizability of spaces of continuous functions 11

1.6. The space K(Ω;E) 14

1.7. Continuous mappings 20

1.8. Continuous extension and restriction 22

1.9. Separation and permutation of variables 23

1.10. Sequential compactness in C_b(Ω;E) 28

Chapter 2. Differentiable Functions 31

2.1. Differentiability 31

2.2. Finite increment theorem 34

2.3. Partial derivatives 37

2.4. Higher order partial derivatives 40

2.5. Spaces C^m(Ω;E), C^m_b (Ω;E), C^m_K(Ω;E), C^m_b (Ω;E) and K^m(Ω;E) 42

2.6. Comparison and metrizability of spaces of differentiable functions 45

2.7. Filtering properties of spaces of differentiable functions 47

2.8. Sequential completeness of spaces of differentiable functions 49

2.9. The space C^m(Ω;E) and the set C^m(Ω;U) 52

Chapter 3. Differentiating Composite Functions and Others 55

3.1. Image under a linear mapping 55

3.2. Image under a multilinear mapping: Leibniz rule 59

3.3. Dual formula of the Leibniz rule 63

3.4. Continuity of the image under a multilinear mapping 65

3.5. Change of variables in a derivative 69

3.6. Differentiation with respect to a separated variable 72

3.7. Image under a differentiable mapping 73

3.8. Differentiation and translation 77

3.9. Localizing functions 79

Chapter 4. Integrating Uniformly Continuous Functions 83

4.1. Measure of an open subset of ℝ^d 83

4.2. Integral of a uniformly continuous function 87

4.3. Case where E is not a Neumann space 92

4.4. Properties of the integral 93

4.5. Dependence of the integral on the domain of integration 96

4.6. Additivity with respect to the domain of integration 99

4.7. Continuity of the integral 101

4.8. Differentiating under the integral sign 103

Chapter 5. Properties of the Measure of an Open Set 105

5.1. Additivity of the measure 105

5.2. Negligible sets 107

5.3. Determinant of d vectors 112

5.4. Measure of a parallelepiped 115

Chapter 6. Additional Properties of the Integral 119

6.1. Contribution of a negligible set to the integral 119

6.2. Integration and differentiation in one dimension 120

6.3. Integration of a function of functions 123

6.4. Integrating a function of multiple variables 125

6.5. Integration between graphs 130

6.6. Integration by parts and weak vanishing condition for a function 133

6.7. Change of variables in an integral 135

6.8. Some particular changes of variables in an integral 142

Chapter 7. Weighting and Regularization of Functions 147

7.1. Weighting 147

7.2. Properties of weighting 150

7.3. Weighting of differentiable functions 153

7.4. Local regularization 157

7.5. Global regularization 162

7.6. Partition of unity 166

7.7. Separability of K∞(Ω) 170

Chapter 8. Line Integral of a Vector Field Along a Path 173

8.1. Paths 173

8.2. Line integral of a field along a path 176

8.3. Line integral along a concatenation of paths 181

8.4. Tubular flow and the concentration theorem 183

8.5. Invariance under homotopy of the line integral of a local gradient 186

Chapter 9. Primitives of Continuous Functions 191

9.1. Explicit primitive of a field with line integral zero 191

9.2. Primitive of a field orthogonal to the divergence-free test fields 194

9.3. Gluing of local primitives on a simply connected open set 195

9.4. Explicit primitive on a star-shaped set: Poincaré’s theorem 197

9.5. Explicit primitive under the weak Poincaré condition 199

9.6. Primitives on a simply connected open set 203

9.7. Comparison of the existence conditions for a primitive 205

9.8. Fields with local primitives but no global primitive 208

9.9. Uniqueness of primitives 210

9.10. Continuous primitive mapping 211

Chapter 10. Additional Results: Integration on a Sphere 213

10.1. Surface integration on a sphere 213

10.2. Properties of the integral on a sphere 215

10.3. Radial calculation of integrals 218

10.4. Surface integral as an integral of dimension d − 1 220

10.5. A Stokes formula 224

Appendix 227

Bibliography 239

Index 243

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