Flux at Infinity of Subharmonic Functions on \(\mathbb{R}^2\)

Abstract

For a $C^2$-function $f(x)$ on a bounded domain $\omega$ in $\mathbb R^2$ the flux is defined by means of outer normal derivative of $f$. In this paper,  we introduce the notion of $flux(f)$ for any real-valued function on $\mathbb R^2$. We define flux on bounded domain $\omega$ and take limits when $\omega$ grows into $\mathbb R^2$ and the limit is defined as "at infinity", the $flux(f)$ at infinity denoted as $flux_\infty f$. This limit $flux_\infty f$ may or may not be finite. The related development is carried out by employing the notion of inversion on $\mathbb R^2$ and the fact that a harmonic function defined outside a compact set in $\mathbb R^2$ is the difference of two subharmonic functions on $\mathbb R^2$ that are harmonic outside a compact set.