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Incomplete lattice sets that control the behaviour of entire harmonic functions

Published online by Cambridge University Press:  17 January 2001

DAVID H. ARMITAGE ,
STEPHEN J. GARDINER ,
WERNER HAUSSMANN  and
LOTHAR ROGGE
DAVID H. ARMITAGE
Affiliation:
Department of Pure Mathematics, Queen's University, Belfast BT7 1NN, Northern Ireland
STEPHEN J. GARDINER
Affiliation:
Department of Mathematics, University College Dublin, Dublin 4, Ireland
WERNER HAUSSMANN
Affiliation:
Department of Mathematics, Gerhard–Mercator–University, 47048 Duisburg, Germany
LOTHAR ROGGE
Affiliation:
Department of Mathematics, Gerhard–Mercator–University, 47048 Duisburg, Germany
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Abstract

Let h be a harmonic function on ℝn of suitably restricted growth. It is known that if h vanishes, or is bounded, on the lattice ℤn−1 × {0}, then the same is true on ℝn−1 × {0}. This paper presents sharp results which show that, if n [ges ] 3, then the same conclusions can be drawn even if information about h is missing on a substantial proportion of the lattice points. As corollaries we obtain uniqueness and Liouville-type theorems for harmonic, and also polyharmonic, functions which improve results by several authors.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 129 , Issue 3 , November 2000 , pp. 549 - 563
Copyright
© 2000 Cambridge Philosophical Society

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