Schwarz reflexion principle in 3-space

Abstract

The Schwarz reflexion principle is well-known in the theory of harmonic functions in a plane. In the three dimensional euclidean space ( = 3-space), however, it seems that some problems remain to be discussed. In this paper, we shall show that any harmonic function h, defined in a domain D within an open ball Fand having vanishing normal derivative on a part E of ΘDfΛdV, can always be continued across E but in general only radially. J. W. Green [2] treated the case where D coincides with V. He showed that h is continued harmonically through E to the entire outside of V if and