Quaternionic Analysis and Transmission Problems

Abstract

It will be considered a transmission problem for the Laplace equation in a bounded domain G ⊂ R³ with a smooth boundary and k inclusions G _i. We find an integral representation of the solution using quaternionic analysis. The unknown densities in this integral representation are determined by a system of integral equations on the boundaries of the inclusions. We show that this system is equivalent to a generalized Riemann-Hilbert transmission problem for a function which is H-regular outside of all boundaries of the domain and of the inclusions. This problem will be solved. Using these results an explicit representation formula for the solution of the transmission problem will be given. At the end there is a short discussion of a transmission problem for the Lame’ system.