Abstract:
This paper deals with an interpretation of the Order Completion Method for systems of nonlinear partial differential equations
(PDEs) in terms of suitable differential algebras of generalized functions.
In particular, it is shown that certain spaces of generalized functions
that appear in the Order Completion Method may be represented as
differential algebras of generalized functions. This result is based on a
characterization of order convergence of sequences of normal lower semi-
continuous functions in terms of pointwise convergence of such sequences.
It is further shown how the mentioned differential algebras are related to
the nowhere dense algebras introduced by Rosinger, and the almost everywhere algebras considered by Verneave, thus unifying two seemingly
different theories of generalized functions. Existence results for generalized solutions of large classes of nonlinear PDEs obtained through the
Order Completion Method are interpreted in the context of the earlier
nowhere dense and almost everywhere algebras.
