Electronic ISBN:  9781470403225 
Product Code:  MEMO/153/729.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 153; 2001; 93 ppMSC: Primary 46; Secondary 26; 35;
In part 1 we construct a diffeomorphism invariant (Colombeautype) 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 Colombeautype generalized functions in the range between the diffeomorphisminvariant 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.ReadershipGraduate 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 Jformalism

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. Noninjectivity 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


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In part 1 we construct a diffeomorphism invariant (Colombeautype) 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 Colombeautype generalized functions in the range between the diffeomorphisminvariant 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.
Graduate students and research mathematicians interested in functional analysis.

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 Jformalism

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. Noninjectivity 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