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
Author(s): Jacques Simon
Series: Analysis for PDEs Set, 3
Edition: 1
Publisher: Wiley, ISTE
Year: 2022
Language: English
Pages: 377
Tags: Semi-Norm, Test Functions, Distributions, Weighting, Regularization, Potentials
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
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