The completion of uniform convergence spaces and an application to nonlinear PDEs. (English) Zbl 1180.35171

van der Walt, Jan Harm

Quaest. Math. 32, No. 3, 371-395 (2009).

Summary: 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.

MSC:

35G50	Systems of nonlinear higher-order PDEs
54E15	Uniform structures and generalizations
54A20	Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
06F30	Ordered topological structures
35A01	Existence problems for PDEs: global existence, local existence, non-existence

Keywords:

uniform convergence spaces; nonlinear partial differential equations; structure and regularity properties