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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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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