Hilbert-space methods in elliptic partial differential equations
Edward M. Landesman · Pacific Journal of Mathematics · 1967
The purpose of this paper is to study, together with applications, those aspects of the theory of Hubert Space which are pertinent to the theory of elliptic partial differential equations.This involves the study of an unbounded operator A from one Hubert Space to another together with its adjoint A*, its pseudo-inverse or generalized reciprocal A" 1 , and its ^-reciprocal A 1 -A*~ι.In order to carry out the results, further properties of the operators A~ι 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 Hubert Space.This Hubert 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 ^f~h 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 G k .Simple examples which illustrate the above theory are provided.1* Preliminaries* We will mainly be interested in Hίlbert Spaces.