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.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 10, pp. 1367–1378, October, 2014.
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Qiao, L. Dirichlet Problems for Harmonic Functions in Half Spaces. Ukr Math J 66, 1530–1543 (2015). https://doi.org/10.1007/s11253-015-1029-9
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DOI: https://doi.org/10.1007/s11253-015-1029-9