Distributions

Description

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. This book is the third of seven volumes dedicated to solving partial dif- ferential equations in physics: Volume 1: Banach, Frechet, Hilbert and Neumann Spaces Volume 2: Continuous Functions Volume 3: Distributions Volume 4: Integration Volume 5: Sobolev Spaces Volume 6: Traces Volume 7: Partial Differential Equations Cover Half-Title Page Title Page Copyright Page Contents Introduction Notations Chapter 1. Semi-Normed Spaces and Function Spaces 1.1. Semi-normed spaces 1.2. Comparison of semi-normed spaces 1.3. Continuous mappings 1.4. Differentiable functions 1.5. Spaces Cm(Ω;E), Cmb(Ω;E) and Cmb(Ω;E) 1.6. Integral of a uniformly continuous function Chapter 2. Space of Test Functions 2.1. Functions with compact support 2.2. Compactness in their whole of support of functions 2.3. The space D(Ω) 2.4. Sequential completeness of D(Ω) 2.5. Comparison of D(Ω) to various spaces 2.6. Convergent sequences in D(Ω) 2.7. Covering by crown-shaped sets and partitions of unity 2.8. Control of the CmK(Ω)-norms by the semi-norms of D(Ω) 2.9. Semi-norms that are continuous on all the C∞K(Ω) Chapter 3. Space of Distributions 3.1. The space D'(Ω;E) 3.2. Characterization of distributions 3.3. Inclusion of C(Ω;E) into D'(Ω;E) 3.4. The case where E is not a Neumann space 3.5. Measures 3.6. Continuous functions and measures Chapter 4. Extraction of Convergent Subsequences 4.1. Bounded subsets of D'(Ω;E) 4.2. Convergence in D'(Ω;E) 4.3. Sequential completeness of D'(Ω;E) 4.4. Sequential compactness in D'(Ω;E) 4.5. Change of the space E of values 4.6. The space E-weak 4.7. The space D'(Ω;E-weak) and extractability Chapter 5. Operations on Distributions 5.1. Distributions fields 5.2. Derivatives of a distribution 5.3. Image under a linear mapping 5.4. Product with a regular function 5.5. Change of variables 5.6. Some particular changes of variables 5.7. Positive distributions 5.8. Distributions with values in a product space Chapter 6. Restriction, Gluing and Support 6.1. Restriction 6.2. Additivity with respect to the domain 6.3. Local character 6.4. Localization-extension 6.5. Gluing 6.6. Annihilation domain and support 6.7. Properties of the annihilation domain and support 6.8. The space D'K(Ω;E) Chapter 7. Weighting 7.1. Weighting by a regular function 7.2. Regularizing character of the weighting by a regular function 7.3. Derivatives and support of distributions weighted by a regular weight 7.4. Continuity of the weighting by a regular function 7.5. Weighting by a distribution 7.6. Comparison of the definitions of weighting 7.7. Continuity of the weighting by a distribution 7.8. Derivatives and support of a weighted distribution 7.9. Miscellanous properties of weighting Chapter 8. Regularization and Applications 8.1. Local regularization 8.2. Properties of local approximations 8.3. Global regularization 8.4. Convergence of global approximations 8.5. Properties of global approximations 8.6. Commutativity and associativity of weighting 8.7. Uniform convergence of sequences of distributions Chapter 9. Potentials and Singular Functions 9.1. Surface integral over a sphere 9.2. Distribution associated with a singular function 9.3. Derivatives of a distribution associated with a singular function 9.4. Elementary Newtonian potential 9.5. Newtonian potential of order n 9.6. Localized potential 9.7. Dirac mass as derivatives of continuous functions 9.8. Heaviside potential 9.9. Weighting by a singular weight Chapter 10. Line Integral of a Continuous Field 10.1. Line integral along a C1 path 10.2. Change of variable in a path 10.3. Line integral along a piecewise C1 path 10.4. The homotopy invariance theorem 10.5. Connectedness and simply connectedness Chapter 11. Primitives of Functions 11.1. Primitive of a function field with a zero line integral 11.2. Tubular flows and concentration theorem 11.3. The orthogonality theorem for functions 11.4. Poincaré’s theorem Chapter 12. Properties of Primitives of Distributions 12.1. Representation by derivatives 12.2. Distribution whose derivatives are zero or continuous 12.3. Uniqueness of a primitive 12.4. Locally explicit primitive 12.5. Continuous primitive mapping 12.6. Harmonic distributions, distributions with a continuous Laplacian Chapter 13. Existence of Primitives 13.1. Peripheral gluing 13.2. Reduction to the function case 13.3. The orthogonality theorem 13.4. Poincaré’s generalized theorem 13.5. Current of an incompressible two dimensional field 13.6. Global versus local primitives 13.7. Comparison of the existence conditions of a primitive 13.8. Limits of gradients Chapter 14. Distributions of Distributions 14.1. Characterization 14.2. Bounded sets 14.3. Convergent sequences 14.4. Extraction of convergent subsequences 14.5. Change of the space of values 14.6. Distributions of distributions with values in E-weak Chapter 15. Separation of Variables 15.1. Tensor products of test functions 15.2. Decomposition of test functions on a product of sets 15.3. The tensorial control theorem 15.4. Separation of variables 15.5. The kernel theorem 15.6. Regrouping of variables 15.7. Permutation of variables Chapter 16. Banach Space Valued Distributions 16.1. Finite order distributions 16.2. Weighting of a finite order distribution 16.3. Finite order distribution as derivatives of continuous functions 16.4. Finite order distribution as derivative of a single function 16.5. Distributions in a Banach space as derivatives of functions 16.6. Non-representability of distributions with values in a Fréchet space 16.7. Extendability of distributions with values in a Banach space 16.8. Cancellation of distributions with values in a Banach space Appendix A.1. Notation and numbering A.2. Semi-normed spaces A.3. Continuous mappings, duality A.4. Continuous or differentiable functions A.5. Integration of uniformly continuous functions Bibliography Index Other titles from iSTE in Mathematics and Statistics