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

 

A uniqueness theorem for harmonic functions


Author: N. V. Rao
Journal: Proc. Amer. Math. Soc. 126 (1998), 1721-1724
MSC (1991): Primary 31A05
MathSciNet review: 1443851
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Abstract: The main result of this note is the following theorem:

Theorem 1. Let $D=\{(x,t); |x|^{2}+t^{2}\leq r^{2},\;t>0\}$ be a half ball in $R^{n+1}$ and $x\in R^{n}$. Assume that $u$ is $C^{1}$ in $\overline D$ and harmonic in $D$, and that for every positive integer $N$ there exists a constant $C_{N}$ such that

\begin{equation*}|\nabla u(x,0)|\leq C_{N}|x|^{N}\quad \hbox { in a neighbourhood $V$ of the origin in $\partial D$;}\tag{1}\end{equation*}

\begin{equation*}u(x,0)\geq u(0,0)\quad \text{\rm in}\quad V.\tag{2}\end{equation*}

Then $u\equiv u(0,0)$.

First we prove it for $R^{2}$, and then we show by induction that it holds for all $n\geq 3$.


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Additional Information

N. V. Rao
Affiliation: Department of Mathematics, University of Toledo, Toledo, Ohio 43606
Email: rnagise@uoft02.utoledo.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04255-5
PII: S 0002-9939(98)04255-5
Received by editor(s): August 20, 1996
Received by editor(s) in revised form: November 20, 1996
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1998 American Mathematical Society