On Dirichlet’s Boundary Value Problem for Certain Anisotropic Differential and Pseudo-Differential Operators

Abstract

In our previous paper [1] a calculus for pseudo-differential operators was used to find periodic solutions for a large class of not necessarily elliptic pseudo-differential equations with constant coefficients. In this paper we consider a generalized homogeneous Dirichlet-problem for pseudo — differential operators with constant coefficients and prove Fredholm’s alternative theorem. We give two examples how to apply this result. The first one is a differential operator arising in stochastics. In the second example we deal with an anisotropic pseudo-differential operator and using the theory of Dirichlet forms (see [4]) it follows that this operator generates a symmetric Hunt process. Some of our results had been already used in [6].