Can There Be a General Nonlinear PDE Theory for Existence of Solutions?

Abstract

Contrary to widespread perception, there is ever since 1994 a unified, general type independent theory for the existence of solutions for very large classes of systems of PDEs. This solution method is based on the Dedekind order completion of suitable spaces of piece-wise smooth functions on the Euclidean domains of definition of the respective PDEs. The method can also deal with associated initial and/or boundary value problems. The solutions obtained can be assimilated with usual measurable functions or even with Hausdorff continuous functions on the respective Euclidean domains. It is important to note that the use of the order completion method does not require any monotonicity condition on the systems of PDEs involved. One of the major advantages of the order completion method is that it eliminates the algebra based dichotomy linear versus nonlinear PDEs, treating both cases with equal ease. Furthermore, the order completion method does not introduce the dichotomy monotonous versus non-monotonous PDEs. None of the known functional analytic methods can exhibit such a performance, since in addition to topology, such methods are significantly based on algebra.