This question is motivated by similar considerations for the Kohn-Laplacian in the Heisenberg group, but it seems that I cannot even give an answer in the Euclidean case, so here we go. 

Suppose that I consider the differential operator (in polar coordinates)
\begin{equation}
D:=\frac{\partial^2}{\partial r^2}{\color{red} -}\frac{1}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial \theta^2}.
\end{equation}
Writing the Laplacian, $\Delta $ in polar coordinates we see in fact that 
\begin{equation} 
D+\frac{2}{r}\frac{\partial}{\partial r }=\Delta.
\end{equation}
For the Laplacian it is well known that the WMP (Weak Maximum Principle)  holds (even the strong one), and since $D$ is a perturbation of the Laplacian by a differential operator of first order, we expect that the WMP to hold for $D$ as well, the problem being that the coefficient of $\frac{\partial}{\partial r}$ has a singularity. 

So, is there something we can say about the WMP for $D$ ?
(Of course assume we work with $C^\infty$ solutions, in a connected domain with $C^\infty$ boundary, etc.)