Hilbert-space methods in elliptic partial differential equations

Abstract

The purpose of this paper is to study, together withapplications, those aspects of the theory of Hubert Spacewhich are pertinent to the theory of elliptic partial differentialequations. This involves the study of an unbounded operatorA from one Hubert Space to another together with its adjointA*, its pseudo-inverse or generalized reciprocal A−, and its*-reciprocal A' =A*−1. In order to carry out the results, further properties of the operators A−1 and A' are developedin this paper.The concept of ellipticity of a partial differential operators introduced via the properties of an operator in a suitably chosenHubert Space. This Hubert Space is the one defined by theoperator Gk, that is, the operator which maps a function intoitself and its first k derivatives. It is shown that ellipticoperators are those that behave in a topological sense thesame as closed and dense restrictions of the operator Gk.Several other characterizations of elliptic operators and givenand their relation to each other is explained. This approachyields existence theorems for strong solutions of elliptic partialdifferential equations and provides methods for gaining strongsolutions from weak solutions.The H−k spaces that arise from the so-called negativenorms and that have been used effectively by several authorsin the study of elliptic partial differential equations areobtained by the use of the ^-reciprocal of the operator Gk.Simple examples which illustrate the above theory areprovided. © 1967 by Pacific Journal of Mathematics.