Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 107 / 1 / i

Let $M={\sup }_{\Omega }u$ and suppose $u(y)=M$. The set ${\Omega }_M=\{x\in \Omega :u(x)=M\}$ is closed in $\Omega$ by continuity. If $x\in {\Omega }_M$, choose a ball $B(x,r)⋐\Omega$. The mean value property for harmonic functions assumed in the question givesThe nonnegative continuous function $M-u$ consequently has integral zero and therefore vanishes throughout the ball. Thus ${\Omega }_M$ is also open. Since $\Omega$ is connected and ${\Omega }_M$ is nonempty, ${\Omega }_M=\Omega$, so $u$ is constant. Applying the same argument to $-u$ proves the assertion for an attained infimum.