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The purpose of this paper is to
study, together with applications, those aspects of the theory of Hilbert Space
which are pertinent to the theory of elliptic partial differential equations.
This involves the study of an unbounded operator A from one Hilbert Space
to another together with its adjoint A∗, its pseudo-inverse or generalized
reciprocal A−1, and its ∗-reciprocal A′ = A∗−1. In order to carry out the
results, further properties of the operators A−1 and A′ are developed in this
paper.
The concept of ellipticity of a partial differential operator is introduced via the
properties of an operator in a suitably chosen Hilbert Space. This Hilbert
Space is the one defined by the operator Gk, that is, the operator which
maps a function into itself and its first k derivatives. It is shown that elliptic
operators are those that behave in a topological sense the same as closed
and dense restrictions of the operator Gk. Several other characterizations of
elliptic operators and given and their relation to each other is explained. This
approach yields existence theorems for strong solutions of elliptic partial
differential equations and provides methods for gaining strong solutions from weak
solutions.
The ℋ−k spaces that arise from the so-called negative norms and that
have been used effectively by several authors in the study of elliptic partial
differential equations are obtained by the use of the *-reciprocal of the operator
Gk.
Simple examples which illustrate the above theory are provided.
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