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.
Table of Contents:
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 Cm (Ω; E), Cmb (Ω; E) and Cmb (Ω; 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(Ω) 21
2.4. Sequential completeness of D(Ω) 24
2.5. Comparison of D(Ω) to various spaces 26
2.6. Convergent sequences in D(Ω) 28
2.7. Covering by crown-shaped sets and partitions of unity 33
2.8. Control of the CK m (Ω)-norms by the semi-norms of D(Ω) 35
2.9. Semi-norms that are continuous on all the CK ∞ (Ω) 38
Chapter 3 Space of Distributions 41
3.1. The space D ′ (Ω; E) 41
3.2. Characterization of distributions 46
3.3. Inclusion of C(Ω; E) into D ′ (Ω; 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 ′ (Ω; E) 65
4.2. Convergence in D ′ (Ω; E) 67
4.3. Sequential completeness of D ′ (Ω; E) 69
4.4. Sequential compactness in D ′ (Ω; E) 71
4.5. Change of the space E of values 74
4.6. The space E-weak 76
4.7. The space D ′ (Ω; 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 ′ (Ω; 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 191
9.2. Distribution associated with a singular function 193
9.3. Derivatives of a distribution associated with a singular function 196
9.4. Elementary Newtonian potential 197
9.5. Newtonian potential of order n 201
9.6. Localized potential 208
9.7. Dirac mass as derivatives of continuous functions 210
9.8. Heaviside potential 214
9.9. Weighting by a singular weight 217
Chapter 10 Line Integral of a Continuous Field 221
10.1. Line integral along a C1 path 221
10.2. Change of variable in a path 225
10.3. Line integral along a piecewise C1 path 228
10.4. The homotopy invariance theorem 231
10.5. Connectedness and simply connectedness 235
Chapter 11 Primitives of Functions 237
11.1. Primitive of a function field with a zero line integral 237
11.2. Tubular flows and concentration theorem 239
11.3. The orthogonality theorem for functions 243
11.4. Poincaré’s theorem 244
Chapter 12 Properties of Primitives of Distributions 247
12.1. Representation by derivatives 247
12.2. Distribution whose derivatives are zero or continuous 251
12.3. Uniqueness of a primitive 253
12.4. Locally explicit primitive 254
12.5. Continuous primitive mapping 256
12.6. Harmonic distributions, distributions with a continuous Laplacian 261
Chapter 13 Existence of Primitives 265
13.1. Peripheral gluing 266
13.2. Reduction to the function case 268
13.3. The orthogonality theorem 270
13.4. Poincaré’s generalized theorem 274
13.5. Current of an incompressible two dimensional field 277
13.6. Global versus local primitives 279
13.7. Comparison of the existence conditions of a primitive 282
13.8. Limits of gradients 283
Chapter 14 Distributions of Distributions 285
14.1. Characterization 285
14.2. Bounded sets 288
14.3. Convergent sequences 289
14.4. Extraction of convergent subsequences 293
14.5. Change of the space of values 294
14.6. Distributions of distributions with values in E-weak 295
Chapter 15 Separation of Variables 297
15.1. Tensor products of test functions 297
15.2. Decomposition of test functions on a product of sets 301
15.3. The tensorial control theorem 303
15.4. Separation of variables 309
15.5. The kernel theorem 311
15.6. Regrouping of variables 317
15.7. Permutation of variables 318
Chapter 16 Banach Space Valued Distributions 323
16.1. Finite order distributions 323
16.2. Weighting of a finite order distribution 326
16.3. Finite order distribution as derivatives of continuous functions 328
16.4. Finite order distribution as derivative of a single function 333
16.5. Distributions in a Banach space as derivatives of functions 335
16.6. Non-representability of distributions with values in a Fréchet space 339
16.7. Extendability of distributions with values in a Banach space 342
16.8. Cancellation of distributions with values in a Banach space 347
Appendix 349
Bibliography 367
Index 371
About the Author :
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.
