Computational Methods and Function Theory 2 (2002), No. 1, 229--247
Copyright Heldermann Verlag 2002
Roger W. Barnard
barnard@math.ttu.edu, Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409-1042, U.S.A.
Leah Cole
colel@wbu.edu, Wayland Baptist University, Division of Mathematics and Sciences, Plainview, TX 79072, U.S.A.
Alexander Yu. Solynin
solynin@pdmi.ras.ru, Steklov Institute of Mathematics at St. Petersburg, Russian Academy of Sciences, Fontanka 27, St. Petersburg, 191011, Russia.
For any two points $a_1$ and $a_2$ in an open disk $\Delta$ on the complex sphere $\overline{\mathbb{C}}$, let $L$ be a curve separating $a_1$ from $a_2$ on $\overline{\mathbb{C}}$, which splits $\overline{\mathbb{C}}$ into two complementary regions $B_1\ni a_1$ and $B_2\ni a_2$. Let $l$ be the part of this curve lying in $\bar\Delta$. In this note we study how small the average harmonic measure $$ \frac{1}{2}(\omega(a_1,l,B_1)+\omega(a_2,l,B_2)) $$ can be. This question can be interpreted as a problem on the minimal average temperature at two points of a long cylinder composed of two media separated by a heating membrane each of which contains a reference point.
Keywords: Harmonic measure, module of a quadrilateral, complete elliptic integral.
MSC 2000: 30C85, 33E05.
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