Extremal function for capacity and estimates of QED constants in

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• 作者：Tao Cheng^a ; ^{tcheng@math.ecnu.edu.cn} ; Shanshuang Yang^b ; ^{syang@mathcs.emory.edu}
• 关键词：30C62 ; 30C70
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：14 January 2017
• 年：2017
• 卷：306
• 期：Complete
• 页码：929-957
• 全文大小：487 K
• 卷排序：306

文摘

This paper is devoted to the study of some fundamental problems on modulus and extremal length of curve families, capacity, and n  -harmonic functions in the Euclidean space RnRn. One of the main goals is to establish the existence, uniqueness, and boundary behavior of the extremal function for the conformal capacity cap(A,B;Ω)cap(A,B;Ω) of a capacitor in RnRn. This generalizes some well known results and has its own interests in geometric function theory and potential theory. It is also used as a major ingredient in this paper to establish a sharp upper bound for the quasiextremal distance (or QED) constant M(Ω)M(Ω) of a domain in terms of its local boundary quasiconformal reflection constant H(Ω)H(Ω), a bound conjectured by Shen in the plane. Along the way, several interesting results are established for modulus and extremal length. One of them is a decomposition theorem for the extremal length λ(A,B;Ω)λ(A,B;Ω) of the curve family joining two disjoint continua A and B in a domain Ω.