
On the Foundations of Nonlinear Generalized Functions I and II
American Mathematical Society (Publisher)
Will be published approx. on 30. July 2001
Book
Paperback/Softback
93 pages
978-0-8218-2729-1 (ISBN)
Description
In part 1 we construct a diffeomorphism invariant (Colombeau-type) differential algebra canonically containing the space of distributions in the sense of L. Schwartz. Employing differential calculus in infinite dimensional (convenient) vector spaces, previous attempts in this direction are unified and completed. Several classification results are achieved and applications to nonlinear differential equations involving singularities are given. Part 2 gives a comprehensive analysis of algebras of Colombeau-type generalized functions in the range between the diffeomorphism-invariant quotient algebra ${\mathcal G}^d = {\mathcal E}_M/{\mathcal N}$ introduced in part 1 and Colombeau's original algebra ${\mathcal G}^e$.Three main results are established: first, a simple criterion describing membership in ${\mathcal N}$ (applicable to all types of Colombeau algebras) is given; second, two counterexamples demonstrate that ${\mathcal G}^d$ is not injectively included in ${\mathcal G}^e$; and finally, it is shown that in the range ""between"" ${\mathcal G}^d$ and ${\mathcal G}^e$ only one more construction leads to a diffeomorphism invariant algebra. In analyzing the latter, several classification results essential for obtaining an intrinsic description of ${\mathcal G}^d$ on manifolds are derived.
More details
Series
Language
English
Place of publication
Providence
United States
Target group
College/higher education
Professional and scholarly
Illustrations
bibliography
Weight
205 gr
ISBN-13
978-0-8218-2729-1 (9780821827291)
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Schweitzer Classification
Content
Part 1. On the Foundations of Nonlinear Generalized Functions I: Introduction Notation and terminology Scheme of construction Calculus C- and J-formalism Calculus on $U_\varepsilon(\Omega)$ Construction of a diffeomorphism invariant Colombeau algebra Sheaf properties Separating the basic definition from testing Characterization results Differential equations Part 2. On the Foundations of Nonlinear Generalized Functions II: Introduction to Part 2 A simple condition equivalent to negligibility Some more calculus Non-injectivity of the canonical homomorphism from ${\mathcal G}^d(\Omega)$ into ${\mathcal G}^e(\Omega)$ Classification of smooth Colombeau algebras between ${\mathcal G}^d(\Omega)$ and ${\mathcal G}^e(\Omega)$ The algebra ${\mathcal G}^2$; classification results Concluding remarks Acknowledgments Bibliography.