Annales Academia Scientiarum Fennicre Series A. I. Mathematica Volumen 16, 1991, 95-112 MONOTONE FUNCTIONS AND EXTREMAL FUNCTIONS FOR CONDENSERS IN R, Shanshuang Yang 0. Introduction In this paper, we study some properties of monotone functions and extremal functions for capacities of condensers. After introducing some notations and pre- liminary results in Section 1, we shall give the construction of a monotone function a^nd prove an oscillation lemma. In Section 2 we also prove that a relative quasiex- tremal distance exceptional set with n-dimensional measure zero is removable for monotone ACl-functions. These are generalizations and modifications of some results due to A. Aseev and A. Syöev [AS]. In Section 3, by using the results ob- tained in Section 2 and some results on conformally invariant variational integrals, we prove a general existence and uniqueness theorem for the extremal function of the conformal capacity of a condenser .E and study the boundary behavior of the extremal. The corresponding results for the special case where .R is a ring are due to F.W. Gehring [G2] and G.D. Mostow [M6]. In Section 4 we establish the cor- responding results for the extremal functions of p-capacities of condensers. Some results obtained here are needed to characterize quasiextremal distance domains and null sets for extremal distances in E" (see [Y] for applications). The author expresses his gratitude to Professor F.W. Gehring for suggesting this topic and for his consistent encouragement and advice. The author would also like to thank Professors J. Heinonen and O. Martio for making many valuable suggestions. 1. Notation and preliminary results We use the following notation for Euclidean n-space R" and its one point compactification -R". Given o € R" and 0 < r < oo, we let B"(arr) denote the open n-ball with center r and radius r and S"-'(*,,r) its boundary. We also let alt. . ., e, denote the unit vectors in the directions of the rectangular coordinate axes in -R". This research was supported in part by grants from the U.S. National Science Foundation and the Institut Mittag-Leffier.