Partial Differential Equations

Abstract

The theory of harmonic spaces was mainly established with the aim to generalize and unify results and methods of classical potential theory for application to an extended class of elliptic and parabolic differential equations of second order. Originally, the theory started with a sheaf of vector spaces of real continuous functions on a locally compact space, playing the role of the sheaf of solutions of a partial differential equation. A convergence property, the boundary minimum principle, the local solvability of the Dirichlet problem are supposed to hold. The most important type of a harmonic space is a Bauer space which is introduced in section 1. In our terminology a Bauer space is locally a harmonic space having a base of regular sets. Semi-elliptic differential operators are treated in section 2. In section 3 we present J. M. BONY’s result that a Bauer space whose harmonic functions are smooth is generated by such a differential operator. Section 4 prepares the material (Sobolev spaces, weak solutions, etc.) needed in section 5 to show that elliptic-parabolic differential operators generate Bauer spaces. Besides the deep result of L. HORMANDER on the hypoellipticity of such operators the theory is completely selfcontained. For sake of simplicity we mostly assume that the constant function 1 is harmonic.KeywordsWeak SolutionDifferential OperatorHarmonic FunctionOpen SubsetDirichlet FormThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.