Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 107 / 1 / b / Solution

For $x,z\in B_{\rho }(y)$, the open balls satisfy $B_{\rho }(x)\subset B_{3\rho }(z)\subset B_{4\rho }(y)$. Nonnegativity and the mean value property for harmonic functions giveTaking the supremum over $x$ and the infimum over $z$ proves the Harnack inequality for harmonic functions:This includes a zero infimum, which forces $u=0$ on the smaller ball.