I am reading Renardy - "An Introduction to PDEs" where I am at the maximum principle for elliptic equations. The strong maximum principle states:
Assume $Lu\ge0\ (Lu\le0)$ in $\Omega$ and assume that $u$ is non constant. If $c=0$ then $u$ does not achieve its maximum (minimum) in the interior of $\Omega$. If $c\le 0$, $u$ cannot achieve a non-negative maximum (non-positive minimum) in the interior. Regardless of the sign of $c$, $u$ cannot be zero at an interior maximum (minimum).
The operator $L$ is elliptic and $\Omega$ is the domain. The exact assumptions on the domain and operator will not be given now, but are assumed to hold for my question.
I now want to try it on the equation
$\begin{align} \Delta u(x,y)&=u(x,y)+\sin(xy),\ &(x,y)\in D_0(1)\\ u(x,y)&=1, \ &(x,y)\in\partial D_0(1) \end{align}$
where $D_0(1)$ is the unit disk centered around the origin.
I want to give an estimate of $\sup|u|$ using the strong maximum principle (and possibly the weak one, too).
My attempt:
The requirements for the theorem are satisfied, as $\Omega = D_0(1)$ and $L$ is elliptic. (But this shall not bother us now)
Since it is also required that $u$ is sufficiently smooth (has to be $\mathcal{C}^2$ in the interior) the maximum and supremum coincide.
I obviously have to use the part of the theorem where $c\le0$. But here already the first problem occurs, I do not have $Lu\le 0$ or $Lu\ge 0$.