# On the Foundations of Nonlinear Generalized Functions I and II

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*Michael Grosser; Eva Farkas; Michael Kunzinger; Roland Steinbauer*

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.

#### Table of Contents

# Table of Contents

## On the Foundations of Nonlinear Generalized Functions I and II

- Contents vii8 free
- Abstract ix10 free
- Preface xi12 free
- Part 1. On the Foundations of Nonlinear Generalized Functions I 114 free
- Chapter 1. Introduction 316
- Chapter 2. Notation and Terminology 619
- Chapter 3. Scheme of construction 821
- Chapter 4. Calculus 1124
- Chapter 5. C- and J-formalism 1629
- Chapter 6. Calculus on U[sub(ε)](Ω) 2235
- Chapter 7. Construction of a diffeomorphism invariant Colombeau algebra 2740
- Chapter 8. Sheaf properties 3952
- Chapter 9. Separating the basic definition from testing 4154
- Chapter 10. Characterization results 4356
- Chapter 11. Differential Equations 5265

- Part 2. On the Foundations of Nonlinear Generalized Functions II 5568
- Chapter 12. Introduction to Part 2 5770
- Chapter 13. A simple condition equivalent to negligibility 5871
- Chapter 14. Some more calculus 6174
- Chapter 15. Non-injectivity of the canonical homomorphism from G[sup(d)](Ω) into G[sup(e)](Ω) 6477
- Chapter 16. Classification of smooth Colombeau algebras between G[sup(d)](Ω) and G[sup(e)](Ω) 7487
- Chapter 17. The algebra G[sup(2)];classification results 8295
- Chapter 18. Concluding remarks 90103

- Acknowledgments 91104
- Bibliography 92105