Let $u$ be harmonic ($\Delta u=0$) on the unit circle, if $u\equiv 0$ on $\partial G$ then $u\equiv0$ on $G$
We set $P=-u\frac{\partial u}{\partial y}, Q=u\frac{\partial u}{\partial x}$
Using Green we get $$Q_x-P_y=(\frac{\partial u}{\partial x})^2+u \frac{\partial^2 u}{\partial x^2}+u\frac{\partial^2 u}{\partial y^2}+(\frac{\partial u}{\partial y})^2=(\frac{\partial u}{\partial x})^2+(\frac{\partial u}{\partial y})^2$$
By green
$$\iint_{G}(\frac{\partial u}{\partial x})^2+(\frac{\partial u}{\partial y})^2=\oint_{\partial G}u(-\frac{\partial u}{\partial y}dx+\frac{\partial u}{\partial x}dy)=0$$
how can I an conclude that $u\equiv 0$? must it be on the unit circle?