The completion of uniform convergence spaces and an application to nonlinear PDEs

Abstract

This paper deals with the solution of large classes of systems of nonlinear partial differential equations (PDEs) in spaces of generalized functions that are constructed as the completion of uniform convergence spaces. The existence result for the mentioned systems of equations are obtained as an application of a basic approximation result, which is formulated entirely in terms of usual real valued functions on open subsets of Euclidean n-space. The structure and regularity properties of the solutions are explained in terms of suitable results relating to the structure of the completion of uniform convergence spaces that are defined as initial structures. In this regard, we include also a detailed discussion of the completion of initial uniform convergence spaces in general.