Half Dirichlet problems and decompositions of Poisson kernels

Following the previous study on the unit ball of Delanghe et al, half-Dirichlet problems for the upper-half space are presented and solved. The solutions further lead to decompositions of the Poisson kernels, and the fact that the classical Dirichlet problems may be solved merely by using Cauchy transformation in the respective two contexts. We show that the only domains for which the half-Dirichlet problems are solvable in the same pattern are balls and half-spaces.