Hausdorff continuous solutions of nonlinear PDEs through the order completion method

TL;DR

This paper demonstrates that solutions to nonlinear PDEs, previously shown to be measurable functions, can be further regularized to Hausdorff continuous functions using an order completion method.

Contribution

It introduces an improved regularity result for nonlinear PDE solutions by employing Dedekind order completion to achieve Hausdorff continuity.

Findings

Solutions can be assimilated with Hausdorff continuous functions

Order completion method enhances regularity of PDE solutions

Extends previous measurable solutions to more regular functions

Abstract

It was shown in 1994, in Oberguggenberger & Rosinger, that very large classes of nonlinear PDEs have solutions which can be assimilated with usual measurable functions on the Euclidean domains of definition of the respective equations. In this paper the regularity of these solutions is significantly improved by showing that they can in fact be assimilated with Hausdorff continuous functions. The method of solution of PDEs is based on the Dedekind order completion of spaces of smooth functions which are defined on the domains of the given equations.