Positivity preservation, elliptic boundary value problems, and green's operators

It is well known that positive Green's operators are not necessarily positivity preserving. This result is important, because many physical problems require positivity in their solutions in order to make sense. In this paper we investigate the matter of just how far from being positivity preserving a positive Green's operator can be. In particular, we will see that there exists positive Green's operators that takes some positive functions to functions with negative mean values. We will also identify a broad class of Green's operators that are not necessarily positivity preserving but have properties related to positivity preservation that one expects from positivity preserving Green's operators. Finally, we will compare the results contained in this paper with those that already exist in the literature on the subject.