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Reflection and uniqueness theorems for harmonic functions

Authors:	D H Armitage

Institution:	Department of Pure Mathematics, The Queen's University of Belfast, Belfast BT7 1NN, Northern Ireland

Abstract:	Suppose that 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 has non-negative lower limit at each point of $\tau$ and tends to 0 sufficiently rapidly on the normal to $\tau$ at 0, then 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 -dimensional manifold.

Keywords:	Harmonic function reflection uniqueness continuation

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