Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 107 / 1 / a / Solution

The weak maximum principle for elliptic operators says that if $c\le 0$ andthenIndeed, a positive interior maximum has $Du=0$ and $\Delta u\le 0$, so the differential inequality is incompatible with a strict positive maximum. Applying this argument to a standard strictly perturbed function and then letting the perturbation tend to zero handles equality and proves the weak statement.