Extremal functions for capacities. (English) Zbl 1185.31002

Let \(c_K\) be the extremal function for the variational 2-capacity \(\text{cap}(K)\) of a compact subset \(K\) of the Royden harmonic boundary \(\delta R\) of an open Riemann surface \(R\) relative to an end \(W\) of \(R\).
The author characterises \(c_K\) as the Dirichlet finite harmonic function \(h\) on \(W\) vanishing continuously on the relative boundary \(\partial W\) of \(W\) that satisfies the following three properties: (1) the normal derivative measure \(*dh\) of \(h\) exists on \(\delta R\), with \(*dh\geq 0\) on \(\delta R\); (2) \(*dh= 0\) on \(\delta R\setminus K\); and (3) \(h=1\) quasi-everywhere on \(K\). He announced this result earlier in [Extremal functions for capacities, Abstracts for Lectures at Function Theory Division, Spring Annual Meeting of Math. Soc. Japan, 23-24 (2008)].
As an application of this characterization, he proves that \(\text{hm}(K)\leq\kappa\cdot\text{cap}(K)^{1/2}\) for every compact subset \(K\) of \(\delta R\), where \(\text{hm}(K)\) is the harmonic measure of \(K\) calculated at a fixed point \(a\) in \(W\), and \(\kappa\) is a constant depending only on \(R\), \(W\) and \(a\).


31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
30F25 Ideal boundary theory for Riemann surfaces
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