Minimal harmonic measure on complementary regions
type: Item
instance of: scholarly article
title: Minimal harmonic measure on complementary regions
zbMATH Open document ID: 1026.30022
DOI: 10.1007/BF03321018
author: Roger W. Barnard
author: Leah Cole
author: A. Yu. Solynin
published in: Computational Methods and Function Theory
publication date: 2003-10-22T00:00:00Z
review text: Let \(\omega (z,E,D)\) denote the harmonic measure of the set \(E\subset \partial D\) with respect to the domain \(D\), evaluated at the point \(z\). The authors study the following extremal problem: Let \(\Delta\) be an open disk, \(a_1,a_2\in \Delta\) and \(l\) be a simple arc in \(\Delta\) dividing it into two domains \(D_1, D_2\) with \(a_1\in D_1\) and \(a_2\in D_2\); find \(\inf \frac{1}{2}(\omega (a_1, l,D_1)+\omega (a_1, l,D_1))\), where the infimum is taken over all possible arcs \(l\). By the conformal invariance of harmonic measure, this problem is equivalent to that of finding \[ \Omega(r):=\inf \tfrac{1}{2}(\omega (r, l,D_1)+\omega (-r, l,D_1)), \] where \(0<r<1\) and the infimum is taken over all arcs \(l\) dividing the unit disk \(D\) into two domains \(D_1, D_2\) with \(r\in D_1\) and \(-r\in D_2\). The authors provide a complete solution of the above problem. The extremal configuration may be described as follows: There exists a certain value \(r_o\) of \(r\) (explicitly evaluated in the paper) such that (a) if \(0<r<r_o\), then the extremal configuration is essentially unique and it is degenerate, i.e. \(l=[r,1], D_2=D\setminus l, D_1=\emptyset\); (b) if \(r_o<r<1\), the extremal configuration is unique and symmetric, i.e. \(l=[-i,i]\); (c) if \(r=r_o\), then there are exactly two extremal configurations: the degenerate and the symmetric. An important step towards the above result is the consideration and solution of another extremal problem which is to find \(\Omega(r)\) under the additional constraint that \(\omega(-r,l,D_2)\) is fixed. This problem is solved by the method of extremal length. Another tool used in the paper is the explicit computation of harmonic measures and extremal lengths in terms of complete elliptic integrals and the application of calculus techniques on functions involving complete elliptic integrals and their derivatives.
reviewed by: Dimitrios Betsakos
Mathematics Subject Classification ID: 30C85
Mathematics Subject Classification ID: 33E05
zbMATH DE Number: 1995503
zbMATH Keywords: harmonic measure
zbMATH Keywords: module of a quadrilateral
zbMATH Keywords: complete elliptic integral
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1995503