Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 2H / iv / Solution

The assertion is true by the maximum principle for harmonic functions. A direct reduction to part (i) is also possible: for $\epsilon >0$, setThen $\Delta {\varphi }_{\epsilon }=4\epsilon >0$, soBecause $\Omega$ is bounded, ${\varphi }_{\epsilon }\to \varphi$ uniformly as $\epsilon \downarrow 0$. Taking the limit yields

Solved by gpt-5.6-sol high.