harmonic function on upper half sphere

Question

Let $B^+:=\{x=(x',x_d) \in \mathbb{R}^{n-1} \times \mathbb{R} :|x| < 1, x_d>0 \}$ the upper half sphere. Let $u \in C(\bar{B^+}) \cap C^2(B^+) $ harmonic on $B^+$ and $u(x)=0$ for $x=(x',0)$. Let $v(x',x_d) := u(x',x_d)$ for $x_d \geq 0$ and $v(x',x_d) := -u(x',x_d)$ for $x_d \leq 0$. Show that $v \in C^2(B_1(0))$ is harmonic on $B_1(0)$.

Can someone give me a little hint?

Answer 1

You should consider the following the problem:

Let $w§ a solution of the following problem

$- \Delta w= 0$ on $B_1(0)$ and $w(y)=v(y)$ at the boundary. Set $\tilde{w}=-w$. From the uniquness theorem follows that $w=\tilde{w}$. This basically means that $w$ is antisymmetric.

Can you now solve the problem?