Some properties of Harmonic Functions

Question

1-Let u be a harmonic function on an open set $\Omega\subset\mathbb{R}^d $. Let $a<b<c $ with $b^2 = ac$ such that $B(x_0, c)\subset\Omega$ compactly. $$ \int_{|\omega|=1}u(x_0+a\omega)u(x_0+c\omega)d\omega = \int_{|\omega|=1}u^2(x_0+b\omega)d\omega. $$ 2-Now Assume is harmonic with the boundary conditions $u=\frac{\partial u}{\partial \nu}= 0$ on an open set of $\partial\Omega$. Show that $u = 0$ everywhere.

Answer 1

1- $a<b<c \text{ and }b^2 = ac\implies a= b\cdot\dfrac{b}{c}:=b\alpha$ also $c= \dfrac{b}{\alpha}$. Consider $f: (0,1)\mapsto \mathbb{R}$ with $$f(\alpha) =\int_{|\omega|=1}u(x_0+b\alpha\omega)u(x_0+\frac{b}{\alpha}\omega)d\omega $$

Then $f$ is well defines and smooth with vanishing derivative thanks to Harmonicity of $u$ and the Green identity as follows:

\begin{eqnarray} \frac{d}{dr}\left(\int_{|\omega|=1}u(x_0+r\omega)d\omega.\right)&= & \int_{|\omega|=1} \nabla u(x_0 +r\omega)\cdot w d\omega\\ &=& \frac{1}{r^{n-1}}\int_{|y-x_0|= r} \nabla u(y)\cdot \frac{y}{r}d\sigma(y)\\ &=&\frac{1}{r^{n-1}}\int_{|y-x_0|<r} \Delta u(y) dy =0 \end{eqnarray} where $0<r<c.$ Therefore, $f$ is constant whence $$f(1) = \lim_{\alpha \to 1^-} f(\alpha)= \int_{|\omega|=1}u^2(x_0+b\omega)d\omega.$$