Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 2I / iv / Solution

Apply the weak weak maximum principle for elliptic operators to the harmonic function $\varphi$ and to $-\varphi$. Their boundary maxima are both zero, giving $\varphi \le 0$ and $\varphi \ge 0$ on the closure. Consequently $\varphi \equiv 0$.