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Minimal Harmonic Measure on Complementary Regions

Minimal Harmonic Measure on Complementary Regions For any two points a1 and a2 in an open disk Δ on the complex sphere \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}${\overline C}$\end{document}, let L be a curve separating a1 from a2 on \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}${\overline C}$\end{document}, which splits \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}${\overline C}$\end{document} into two complementary regions B1 э a1 and B2 э a2. Let l be the part of this curve lying in \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}${\bar\Delta}$\end{document}. In this note we study how small the average harmonic measure \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${1\over 2}(\omega(a_{1},\ l,\ B_{1})+\omega(a_{2},\ l,\ B_{2}))$$\end{document} 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Minimal Harmonic Measure on Complementary Regions

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References (18)

Publisher
Springer Journals
Copyright
Copyright © Heldermann  Verlag 2002
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/bf03321018
Publisher site
See Article on Publisher Site

Abstract

For any two points a1 and a2 in an open disk Δ on the complex sphere \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}${\overline C}$\end{document}, let L be a curve separating a1 from a2 on \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}${\overline C}$\end{document}, which splits \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}${\overline C}$\end{document} into two complementary regions B1 э a1 and B2 э a2. Let l be the part of this curve lying in \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}${\bar\Delta}$\end{document}. In this note we study how small the average harmonic measure \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${1\over 2}(\omega(a_{1},\ l,\ B_{1})+\omega(a_{2},\ l,\ B_{2}))$$\end{document} 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.

Journal

Computational Methods and Function Theory – Springer Journals

Published: Jun 1, 2003

Keywords: Harmonic measure; module of a quadrilateral; complete elliptic integral; 30C85; 33E05

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