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

Choose $0<r<\mathrm{dist}(x,\partial \Omega )$. Differentiating the ball mean-value formula with respect to its centre and then using the divergence theorem givesConsequently the interior derivative estimate for a harmonic function yieldsThe constant may depend on $x$ and its distance from the boundary, but it is independent of $u$.