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

• 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

• 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