Pluriharmonics in general potential theories

TL;DR

This paper explores pluriharmonics within general potential theories linked to convex cones, classifying symmetric cases and examining their role in solving the Dirichlet problem.

Contribution

It introduces the concept of edge functions in potential theory, classifies symmetric cases, and connects pluriharmonics to the Dirichlet problem.

Findings

Structural results on pluriharmonics and edges

Classification of symmetric cases

Edge functions' relevance to Dirichlet problem

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.