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.
1. Semi-Normed Spaces and Function Spaces.
2. Space of Test Functions.
3. Space of Distributions.
4. Extraction of Convergent Subsequences.
5. Operations on Distributions.
6. Restriction, Gluing and Support.
8. Regularization and Applications.
9. Potentials and Singular Functions.
10. Line Integral of a Continuous Field.
11. Primitives of Functions.
12. Properties of Primitives of Distributions.
13. Existence of Primitives.
14. Distributions of Distributions.
15. Separation of Variables.
16. Banach Space Valued Distributions.
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.
Table of Contents
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