Flux at Infinity of Subharmonic Functions on \(\mathbb{R}^2\) (European Journal of Pure and Applied Mathematics, 2025)
For a $C^2$-function $f(x)$ on a bounded domain $ω$ 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 $ω$ and take limits when $ω$ grows into $\mathbb R^2$ and the limit is defined as "at infinity", the $flux(f)$ at infinity denoted as $flux_∞ f$. This limit $flux_∞ f$ 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.
Keywords: Subharmonic Function; Harmonic Function; Flux