Separately Subharmonic and Harmonic Functions are Subharmonic

TL;DR

This paper proves that a function which is subharmonic in one variable and harmonic in another is necessarily subharmonic overall, resolving a long-standing open problem in potential theory.

Contribution

It establishes that separate subharmonicity and harmonicity imply joint subharmonicity for functions of two variables, a previously unresolved question.

Findings

Proves that functions subharmonic in one variable and harmonic in another are jointly subharmonic.

Solves a long-standing open problem in potential theory.

Provides a new understanding of the interplay between harmonic and subharmonic functions.

Abstract

We will prove that a function u(x,y) defined on a domain of RpxRq that is subharmonic in one variable and harmonic in the other is (jointly) subharmonic. This solves a long-standing open problem.