Extremal functions for capacities

Abstract

The extremal function $c_K$ for the variational 2-capacity $\mathrm{cap}(K)$ of a compact subset $K$ of the Royden harmonic boundary $\delta R$ of an open Riemann surface $R$ relative to an end $W$ of $R$, referred to as the capacitary function of $K$, is characterized as the Dirichlet finite harmonic function $h$ on $W$ vanishing continuously on the relative boundary $\partial W$ of $W$ satisfying the following three properties: the normal derivative measure $*\text{dh}$ of $h$ exists on $\delta R$ with $*\text{dh}\geqq 0$ on $\delta R$; $*\text{dh}=0$ on $\delta R\setminus K$; $h=1$ quasieverywhere on $K$. As a simple application of the above characterization, we will show the validity of the following inequality

$$\text{hm}\left(K\right)\leqq \kappa •\text{cap}{\left(K\right)}^{1/2}$$

for every compact subset $K$ of $\delta R$, where $\mathrm{hm}(K)$ is the harmonic measure of $K$ calculated at a fixed point $a$ in $W$ and $\kappa$ is a constant depending only upon the triple $(R,W,a)$.