The <weak maximum principle for elliptic operators> fails without a condition at infinity. The function
$$
u(x)=x_n
$$
is harmonic on the upper half-space, continuous on its closure, and vanishes on the boundary, but it is positive and unbounded in the interior.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-107/2/b/solution