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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 $n\geq 3$.

References [Enhancements On Off] (What's this?)

  • [1] M. S. Baouendi and L. P. Rothschild, Unique continuation and a Schwarz reflection principle for analytic sets, Comm. P. D. E. 18(11) (1993), pp. 1961-1970. MR 94i:32014
  • [2] M. S. Baouendi and L. P. Rothschild, A local Hopf lemma and unique continuation for harmonic functions, Int. Math. Res. Notices, 1993, no. 8, pp. 245-251. MR 94i:31008

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Additional Information

N. V. Rao
Affiliation: Department of Mathematics, University of Toledo, Toledo, Ohio 43606

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

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