Extremal function for capacity and estimates of QED constants in [formula omitted

Abstract

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 Rn. 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) of a capacitor in Rn. 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) of a domain in terms of its local boundary quasiconformal reflection constant 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) of the curve family joining two disjoint continua A and B in a domain Ω.