Lecture Notes: Non-Standard Approach to J.F. Colombeau's Theory of Generalized Function: University of Vienna, Austria, May 2006

TL;DR

This paper introduces a non-standard analysis approach to Colombeau's generalized functions, enhancing scalar properties and simplifying the theory, with applications to PDEs, physics, and geometry.

Contribution

It develops a non-standard framework for Colombeau's theory, improving scalar properties and reducing complexity, facilitating broader applications and easier axiomatization.

Findings

Scalars form algebraically closed non-archimedean fields

Simplified axiomatization of generalized function spaces

Applications demonstrated in PDEs and mathematical physics

Abstract

In these lecture notes we present an introduction to non-standard analysis especially written for the community of mathematicians, physicists and engineers who do research on J. F. Colombeau' theory of new generalized functions and its applications. The main purpose of our non-standard approach to Colombeau' theory is the improvement of the properties of the scalars of the varieties of spaces of generalized functions: in our non-standard approach the sets of scalars of the functional spaces always form algebraically closed non-archimedean Cantor complete fields. In contrast, the scalars of the functional spaces in Colombeau's theory are rings with zero divisors. The improvement of the scalars leads to other improvements and simplifications of Colombeau's theory such as reducing the number of quantifiers and possibilities for an axiomatization of the theory. Some of the algebras we construct in these notes have already counterparts in Colombeau's theory, other seems to be without counterpart. We present applications of the theory to PDE and mathematical physics. Although our approach is directed mostly to Colombeau's community, the readers who are already familiar with non-standard methods might also find a short and comfortable way to learn about Colombeau's theory: a new branch of functional analysis which naturally generalizes the Schwartz theory of distributions with numerous applications to partial differential equations, differential geometry, relativity theory and other areas of mathematics and physics.