Extremal functions for capacities

概要

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The extremal function cK for the variational 2-capacity cap(K) of a compact subset K of the Royden harmonic boundary δR 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 δR with *dh≧0 on δR; *dh=0 on δR\\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 δR, 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).
2009-04-01