Past exam of the algebra 1 and differential equations 1 course of the University of Oxford / 2014 / 5 / b / i

Write the elliptic differential operator as $L=\Delta +r{\partial }_x+s{\partial }_y$. At an interior maximum, $u_x=u_y=0$ and $u_{xx}+u_{yy}\le 0$. Therefore $Lu\le 0$ there, contradicting $Lu=f>0$.

The extreme value theorem ensures that a maximum exists on the compact closed disk. It cannot lie in the interior, so the maximum is attained on $\partial D$. The drift terms disappear at a stationary point, which is why the strict-sign proof is so short.