Open Access
April, 2009 Extremal functions for capacities
Mitsuru NAKAI
J. Math. Soc. Japan 61(2): 345-361 (April, 2009). DOI: 10.2969/jmsj/06120345


The extremal function cK for the variational 2-capacity cap(K) of a compact subset K of the Royden harmonic boundary δ 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 W of W satisfying the following three properties: the normal derivative measure *dh of h exists on δ R with *dh 0 on δ R; *dh=0 on δ R K; h=1 quasieverywhere on K. As a simple application of the above characterization, we will show the validity of the following inequality

hm ( K ) κ cap ( K ) 1 / 2

for every compact subset K of δ R, where hm(K) is the harmonic measure of K calculated at a fixed point a in W and κ is a constant depending only upon the triple (R,W,a).


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Mitsuru NAKAI. "Extremal functions for capacities." J. Math. Soc. Japan 61 (2) 345 - 361, April, 2009.


Published: April, 2009
First available in Project Euclid: 13 May 2009

zbMATH: 1185.31002
MathSciNet: MR2532892
Digital Object Identifier: 10.2969/jmsj/06120345

Primary: 31A15
Secondary: 30F20 , 30F25

Keywords: (mutual) Dirichlet integral , Bergman kernel , capacity , Green kernel , harmonic measure , Hilbert space , Neumann kernel , normal derivative measure , reproducing kernel , Royden (harmonic) boundary , Royden compactification

Rights: Copyright © 2009 Mathematical Society of Japan

Vol.61 • No. 2 • April, 2009
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