By setting up appropriate uniform convergence structures, we are able to reformulate the Order Completion Method of Oberguggenberger and Rosinger in a setting that more closely resembles the usual topological constructions for solving PDEs. As an application, we obtain existence and uniqueness results for the solutions of arbitrary continuous, nonlinear PDEs.
Deep Dive into The order completion method for systems of nonlinear PDEs: Pseudo-topological perspectives.
By setting up appropriate uniform convergence structures, we are able to reformulate the Order Completion Method of Oberguggenberger and Rosinger in a setting that more closely resembles the usual topological constructions for solving PDEs. As an application, we obtain existence and uniqueness results for the solutions of arbitrary continuous, nonlinear PDEs.
1.1. General, Type Independent Theories for PDEs It is widely believed that there is no general, type independent theory concerning existence and regularity of solutions to arbitrary nonlinear PDEs. Indeed, the book 'Lecture notes on PDEs' by I. V. Arnold (Arnold, 2004) starts with the following remark: "In contrast to ordinary differential equations, there is no unified theory of partial differential equations. Some equations have their own theories, while others have no theory at all. The reason for this complexity is a more complicated geometry..." Within the confines of functional analysis, and more broadly speaking, general topology, this was until very recently the case. However, within the last fifteen years, two such general, type independent theories for the solutions PDEs appeared.
In 1994 there appeared a type independent theory of existence and regularity of solutions to nonlinear systems of PDEs based on order completion of sets of functions, see (Oberguggenberger and Rosinger, 1994). This theory yields the existence, and uniqueness, of solutions to arbitrary continuous and nonlinear systems of PDEs, with the solutions assimilated with Hausdorff continuous functions, see (Anguelov and Rosinger, 2005).
Also, within the confines of functional analysis, there recently emerged a general and type independent theory of PDEs, see (Neuberger, 1997) through (Neuberger, 2005). This theory, developed by Neuberger, is based on approximations within Hilbert space obtained by a generalized method of Steepest Descent. This method has yielded exceptional numerical results.
Both methods are in fact able to solve equations that are far more general than PDEs. This is precisely the feature which makes them so powerful and type independent when applied to the particular case of PDEs.
Let us recall the general construction for solving a PDE of the form
by topological methods. Here Ω ⊆ R n is supposed to be nonempty and open. The right hand term f is assumed continuous, and the unknown function satisfies, initially at least, u ∈ C m (Ω). The operator T (x, D) is supposed to be defined by some continuous function F : Ω × R M → R through T (x, D) u (x) = F (x, u (x) , …, D α u (x) , …) , x ∈ Ω and |α| ≤ m (2)
The general construction for finding generalized solutions to (1) through (2), within the context of topology, and specifically functional analysis, is summarized as follows:
(1) Start with a PDE-operator T mapping some initial space of classical functions X into some space of functions Y , with the righthand term f ∈ Y . (2) Define some structure, e.g. a uniformity, on Y .
(3) Define a structure of the same sort on X through pullback of the operator T to obtain a space X T so that T is compatible with the customary structures on X and Y .
(4) Construct the completions of X T and Y , say X ♯ T and Y ♯ , and extend T to a mapping
T , one obtains a generalized solution to the equation
The order completion method operates in a similar fashion, see (Oberguggenberger and Rosinger, 1994). It is based on the fundamental approximation result that, under a condition which is necessary for the existence of a classical solution at x ∈ Ω, every continuous righthand term f in (1) can be approximated from below (or from above) by functions in C m nd (Ω) where
In this regard, for any f ∈ C 0 (Ω) we have
One now proceeds as follows: On the set C 0 nd (Ω) one considers the equivalence relation
On the quotient space M 0 (Ω) = C 0 nd (Ω) / ∼, one introduces the partial order
By the continuity of the operator T (x, D) it follows that if u ∈ C m nd (Ω), and u ∈ C m (Ω \ Γ), then T (x, D) u ∈ C 0 (Ω \ Γ) so that one has the well defined mapping
On the set C m nd (Ω) one considers the equivalence relation induced by the PDE operator
With the mapping (7) one can now associate in a canonical way an injective mapping
where
Next one introduces a partial order on M m T (Ω) by
This construction results in an order isomorphic embedding
By extension of T to the Dedekind completion M m T (Ω) ♯ one obtains the following commutative diagram
with T ♯ an order isomorphism. Hence, for every f ∈ C 0 (Ω), the generalized equation
The aim of this paper is to cast the Order Completion Method of Oberguggenberger and Rosinger within the framework of the traditional topological and function analytical method. In order to achieve this aim it is necessary to consider topological type structures that are more general than the usual Hausdorff-Kuratowski-Bourbaki concept of topology. Note that this need for generalized concepts of topology is not unique to the problem considered here, but appears frequently in analysis-even in the relatively simple setting of locally convex linear spaces. Indeed, recall that there is no topology on the topological dual E * of a locally convex space E such that the simple evaluation mapping
is jointly continuous, unless E is a normable space. Other examples include some natural notions of convergence
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