Dirichlet Problems for Harmonic Functions in Half Spaces - Qiao, Lei (Ukrains’kyi Matematychnyi Zhurnal, 2014)
In our paper, we prove that if the positive part u^+(x) of a harmonic function u(x) in a half space satisfies the condition of slow growth, then its negative part u^-(x) can also be dominated by a similar growth condition. Moreover, we give an integral representation of the function u(x). Further, a solution of the Dirichlet problem in the half space for a rapidly growing continuous boundary function is constructed by using the generalized Poisson integral with this boundary function.