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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Reflection and uniqueness theorems
for harmonic functions


Author: D. H. Armitage
Journal: Proc. Amer. Math. Soc. 128 (2000), 85-92
MSC (1991): Primary 31B05
Published electronically: June 24, 1999
MathSciNet review: 1622753
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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.


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Additional Information

D. H. Armitage
Affiliation: Department of Pure Mathematics, The Queen’s University of Belfast, Belfast BT7 1NN, Northern Ireland
Email: d.armitage@qub.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04994-1
PII: S 0002-9939(99)04994-1
Keywords: Harmonic function, reflection, uniqueness, continuation
Received by editor(s): February 7, 1995
Received by editor(s) in revised form: March 4, 1998
Published electronically: June 24, 1999
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1999 American Mathematical Society