Reflection and uniqueness theorems for harmonic functions

Suppose that h is harmonic on an open half-ball beta in R-N such that the origin 0 is the centre of the at part tau of the boundary partial derivative beta. If h has non-negative lower limit at each point of tau and h tends to 0 sufficiently rapidly on the normal to tau at 0, then h has a harmonic continuation by reflection across tau. Under somewhat stronger hypotheses, the conclusion is that h = 0. These results strengthen recent theorems of Baouendi and Rothschild. While the at boundary set tau can be replaced by a spherical surface, it cannot in general be replaced by a smooth (N - 1)-dimensional manifold.