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Reflection and Uniqueness Theorems for Harmonic Functions

D. H. Armitage
Proceedings of the American Mathematical Society
Vol. 128, No. 1 (Jan., 2000), pp. 85-92
Stable URL: http://www.jstor.org/stable/119387
Page Count: 8
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Reflection and Uniqueness Theorems for Harmonic Functions
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Abstract

Suppose that h is harmonic on an open half-ball β in RN such that the origin 0 is the centre of the flat part τ of the boundary ∂ β . If h has non-negative lower limit at each point of τ and h tends to 0 sufficiently rapidly on the normal to τ at 0, then h has a harmonic continuation by reflection across τ . 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 τ can be replaced by a spherical surface, it cannot in general be replaced by a smooth (N - 1)-dimensional manifold.

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