No. A positive zeroth-order coefficient can overturn the <weak maximum principle for elliptic operators>. On $\Omega=(0,1)$ take
$$
L=\frac{d^2}{dx^2}+\pi^2,
\qquad
u(x)=\sin(\pi x).
$$
Then $L\nu=0$, $\nu=0$ on $\partial\Omega$, and $\nu>0$ in $\Omega$. Thus the boundary maximum is zero while the interior maximum is one. This is a counterexample satisfying uniform ellipticity and zero drift.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 107 / 2 / iii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-107/2/iii/solution