Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 107 / 2 / c / Solution

For $n\ge 3$, the radial functionis harmonic on ${\mathbb{R}}^n\setminus \overline{B_1(0)}$, vanishes on the unit sphere, and is positive in the domain. It therefore violates the weak maximum principle for elliptic operators. In two dimensions the corresponding counterexample is $u(x)=\log |x|$, since the fundamental solution of the Laplace equation changes from a power to a logarithm.