Distributions

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.