A uniqueness theorem for harmonic functions

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

1. $|\nabla u(x,0)| \leq C_{N} |x|^{N}$ in a neighbourhood $V$ of the origin in $\partial D$;
2. $u(x,0)\geq u(0,0)$ in $V$.

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

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