The <weak maximum principle for elliptic operators> says that if $c\leq0$ and
$$
\Delta u+cu\geq0,
$$
then
$$
\boxed{\sup_\Omega u\leq\max\{0,\sup_{\partial\Omega}u\}.}
$$
Indeed, a positive interior maximum has $Du=0$ and $\Delta u\leq0$, 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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-107/1/a/solution