Reflection and uniqueness theorems for harmonic functions

Abstract:

Suppose that $h$ is harmonic on an open half-ball $\beta$ in $R^{N}$ such that the origin 0 is the centre of the flat part $\tau$ of the boundary $\partial \beta$. 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 $h\equiv 0$. These results strengthen recent theorems of Baouendi and Rothschild. While the flat boundary set $\tau$ can be replaced by a spherical surface, it cannot in general be replaced by a smooth $(N-1)$-dimensional manifold.