A Discontinuous Differential Calculus in the Framework Colombeau's Full Algebra

TL;DR

This paper develops a new differential calculus within Colombeau's full algebra framework, extending classical theorems and exploring applications to differential equations in totally disconnected spaces.

Contribution

It introduces a novel differential calculus for functions between totally disconnected spaces using Colombeau's full algebra, including generalized pointvalues and theorems like Embedding and Open Mapping.

Findings

Embedding Theorem holds in this framework

Open Mapping Theorem holds in this framework

Applications to differential equations are demonstrated

Abstract

Starting from the Colombeau's full generalized functions, the sharp topologies and the notion of generalized points, we introduce a new kind differential calculus (for functions between totally disconnected spaces). We study generalized pointvalues, Colombeau's differential algebra, holomorphic and analytic functions. We show that the Embedding Theorem and the Open Mapping Theorem hold in this framework. Moreover, we study some applications in differential equations.