# On the foundations of nonlinear generalized functions I and II

### About this Title

**Michael Grosser**, **Eva Farkas**, **Michael Kunzinger** and **Roland Steinbauer**

Publication: Memoirs of the American Mathematical Society

Publication Year
2001: Volume 153, Number 729

ISBNs: 978-0-8218-2729-1 (print); 978-1-4704-0322-5 (online)

DOI: http://dx.doi.org/10.1090/memo/0729

MathSciNet review: 1848157

MSC: Primary 46F30; Secondary 26E15, 35A08, 35D05, 46G05, 46T30

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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 ${\mathcal
G}^e$. 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.

Readership

Graduate students and research mathematicians interested in
functional analysis.

### Table of Contents

**Chapters**

- Part 1. On the foundations of nonlinear generalized functions I
- 1. Introduction
- 2. Notation and terminology
- 3. Scheme of construction
- 4. Calculus
- 5. C- and J-formalism
- 6. Calculus on $U_\epsilon (\Omega )$
- 7. Construction of a diffeomorphism invariant Colombeau algebra
- 8. Sheaf properties
- 9. Separating the basic definition from testing
- 10. Characterization results
- 11. Differential equations
- Part 2. On the foundations of nonlinear generalized functions II
- 12. Introduction to Part 2
- 13. A simple condition equivalent to negligibility
- 14. Some more calculus
- 15. Non-injectivity of the canonical homomorphism from $\mathcal
{G}^d(\Omega )$ into $\mathcal {G}^e(\Omega )$
- 16. Classification of smooth Colombeau algebras between $\mathcal
{G}^d(\Omega )$ and $\mathcal {G}^e(\Omega )$
- 17. The algebra $\mathcal {G}^2$; classification results
- 18. Concluding remarks