Vol. 21, No. 1, 1967

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Hilbert-space methods in elliptic partial differential equations

Edward Milton Landesman

Vol. 21 (1967), No. 1, 113–131

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 A1, and its -reciprocal A= A∗−1. In order to carry out the results, further properties of the operators A1 and Aare 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 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.

Mathematical Subject Classification
Primary: 47.65
Received: 15 October 1965
Published: 1 April 1967
Edward Milton Landesman