Jacques Simon

Distributions (eBook, ePUB)

• Format: ePub

Produktbeschreibung

This book presents a simple and original theory of distributions, both real and vector, adapted to the study of partial differential equations. It deals with value distributions in a Neumann space, that is, in which any Cauchy suite converges, which encompasses the Banach and Fréchet spaces and the same "weak" spaces. Alongside the usual operations - derivation, product, variable change, variable separation, restriction, extension and regularization - Distributions presents a new operation: weighting. This operation produces properties similar to those of convolution for distributions defined in any open space. Emphasis is placed on the extraction of convergent sub-sequences, the existence and study of primitives and the representation by gradient or by derivatives of continuous functions. Constructive methods are used to make these tools accessible to students and engineers.

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Produktdetails

• Verlag: John Wiley & Sons
• Seitenzahl: 416
• Erscheinungstermin: 19. August 2022
• Englisch
• ISBN-13: 9781394165353
• Artikelnr.: 65464152

Autorenporträt

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

Inhaltsangabe

Introduction ix

Notations xv

Chapter 1 Semi-Normed Spaces and Function Spaces 1

1.1. Semi-normed spaces 1

1.2. Comparison of semi-normed spaces 4

1.3. Continuous mappings 6

1.4. Differentiable functions 8

1.5. Spaces C¯m (Omega; E), C¯mb (Omega; E) and C¯mb (Omega; E) 11

1.6. Integral of a uniformly continuous function 14

Chapter 2 Space of Test Functions 17

2.1. Functions with compact support 17

2.2. Compactness in their whole of support of functions 19

2.3. The space D(Omega) 21

2.4. Sequential completeness of D(Omega) 24

2.5. Comparison of D(Omega) to various spaces 26

2.6. Convergent sequences in D(Omega) 28

2.7. Covering by crown-shaped sets and partitions of unity 33

2.8. Control of the CK m (Omega)-norms by the semi-norms of D(Omega) 35

2.9. Semi-norms that are continuous on all the CK infinity (Omega) 38

Chapter 3 Space of Distributions 41

3.1. The space D ' (Omega; E) 41

3.2. Characterization of distributions 46

3.3. Inclusion of C(Omega; E) into D ' (Omega; E) 48

3.4. The case where E is not a Neumann space 53

3.5. Measures 57

3.6. Continuous functions and measures 63

Chapter 4 Extraction of Convergent Subsequences 65

4.1. Bounded subsets of D ' (Omega; E) 65

4.2. Convergence in D ' (Omega; E) 67

4.3. Sequential completeness of D ' (Omega; E) 69

4.4. Sequential compactness in D ' (Omega; E) 71

4.5. Change of the space E of values 74

4.6. The space E-weak 76

4.7. The space D ' (Omega; E-weak) and extractability 78

Chapter 5 Operations on Distributions 81

5.1. Distributions fields 81

5.2. Derivatives of a distribution 84

5.3. Image under a linear mapping 91

5.4. Product with a regular function 94

5.5. Change of variables 100

5.6. Some particular changes of variables 107

5.7. Positive distributions 109

5.8. Distributions with values in a product space 113

Chapter 6 Restriction, Gluing and Support 117

6.1. Restriction 117

6.2. Additivity with respect to the domain 121

6.3. Local character 122

6.4. Localization-extension 125

6.5. Gluing 128

6.6. Annihilation domain and support 130

6.7. Properties of the annihilation domain and support 133

6.8. The space DK ' (Omega; E) 137

Chapter 7 Weighting 141

7.1. Weighting by a regular function 141

7.2. Regularizing character of the weighting by a regular function 144

7.3. Derivatives and support of distributions weighted by a regular weight 148

7.4. Continuity of the weighting by a regular function 150

7.5. Weighting by a distribution 153

7.6. Comparison of the definitions of weighting 156

7.7. Continuity of the weighting by a distribution 159

7.8. Derivatives and support of a weighted distribution 161

7.9. Miscellanous properties of weighting 165

Chapter 8 Regularization and Applications 169

8.1. Local regularization 169

8.2. Properties of local approximations 174

8.3. Global regularization 175

8.4. Convergence of global approximations 178

8.5. Properties of global approximations 180

8.6. Commutativity and associativity of weighting 183

8.7. Uniform convergence of sequences of distributions 188

Chapter 9 Potentials and Singular Functions 191

9.1. Surface integral over a sphere