  In 1994 we showed that very large classes of systems of nonlinear PDEs have
solutions which can be assimilated with usual measurable functions on the
Euclidean domains of definition of the respective equations. Recently, the
regularity of such solutions has significantly been 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. In
this way, the method does not use functional analytic approaches, or any of the
customary distributions, hyperfunctions, or other generalized functions.

Analysis of PDEs