On conjugate harmonic functions in Euclidean space

Abstract

Abstract In this paper we consider the problem of constructing in domains Ωof ℝ m +1 with a specific geometric property, a conjugate harmonic V to a given harmonic function U , as a direct generalization of the complex plane case. This construction is carried out in the framework of Clifford analysis which focusses on the so‐called monogenic functions, i.e. null solutions of the Dirac operator. An explicit formula of the associated monogenic function F = U + ē 0 V in terms of a harmonic potential is constructed and the interconnection with the Stein–Weiss notion of conjugate harmonicity will be shown. Copyright © 2002 John Wiley & Sons, Ltd.