## One simple maximum principle

$\mathbf{Problem:}$If $L$ is elliptic with $c<0$ in a bounded domain $\Omega$, and $u\in C^2(\Omega)\cap C^0(\overline{\Omega})$ satisfies $Lu=f$ in $\Omega$, then
$\displaystyle \sup\limits_{\Omega}\leq \sup\limits_{\partial \Omega}|u|+\sup\limits_{\Omega}|f/c|$

$\mathbf{Proof:}$ Let $v=u-\sup\limits_{\Omega}|u|-\sup\limits_{\Omega}|f/c|$

Then $Lv=Lu-c(\sup\limits_{\Omega}|u|-\sup\limits_{\Omega}|f/c|)\geq 0$ and $v(x)\leq 0$ on $\partial \Omega$. By the maximum principle, we conclude, $v\leq 0$
So $u\leq \sup\limits_{\Omega}|u|-\sup\limits_{\Omega}|f/c|$
We also have $-u$ satisfies the above inequality, so the conclusion holds.

$\text{Q.E.D}\hfill \square$

$\mathbf{Remark:}$