Extremal function for capacity and estimates of QED constants in Rⁿ

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.