Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 107 / 1 / a / i / Solution

Fix $x\in \Omega$ and $R<\mathrm{dist}(x,\partial \Omega )$. The mean value property for harmonic functions isTo prove the spherical identity, translate $x$ to the origin and put $F(r)=|S^{n-1}{|}^{-1}{\int }_{S^{n-1}}u(r\theta )d{S}_{\theta }$. The divergence theorem givesHence $F(r)={\lim }_{s\downarrow 0}F(s)=u(0)$. Integrating the spherical identity in polar coordinates gives the ball identity.