Pluriharmonics in general potential theories

Abstract

The general purpose of this paper is to investigate the notion of "pluriharmonics" for the general potential theory associated to a convex cone $F\subset {\rm Sym}^2({\bf R}^n)$. For such $F$ there exists a maximal linear subspace $E\subset F$, called the edge, and $F$ decomposes as $F=E \oplus F_0$. The pluriharmonics or edge functions are $u$'s with $D^2u \in E$. Many subequations $F$ have the same edge $E$, but there is a unique smallest such subequation. These are the focus of this investigation. Structural results are given. Many examples are described, and a classification of highly symmetric cases is given. Finally, the relevance of edge functions to the solutions of the Dirichlet problem is established.