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


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