Hausdorff Continuous Solutions of Nonlinear Partial Differential Equations through the Order Completion Method

Abstract

It was shown in [15] that very large classes of nonlinear partial differential equations (PDE's) 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 improved significantly by showing that they can in fact be assimilated with Hausdorff continuous functions. The method of solution of PDE's is based on the Dedekind order completion of spaces of smooth functions which are defined on the domains of the given equations.