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**Hausdorff continuous solutions of nonlinear partial differential equations through the order completion method.**
*(English)*
Zbl 1330.35080

Summary: It was shown in [M. B. Oberguggenberger and E. E. Rosinger, Solution of continuous nonlinear PDEs through order completion, Amsterdam: Elsevier Science (1994; Zbl 0821.35001)] that very large classes of nonlinear partial differential equations (PDE’s) have solutions which can be assimilated with the 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.