Supremum norm barrier for an elliptic Dirichlet problem
= Supremum norm barrier for an elliptic Dirichlet problem
{title2=$\|u\|_\infty\leq M+e^{2d}F$}
If $c\leq0$, $\Delta u+cu=f$, and $u=\psi$ on the boundary of a bounded domain inside $\{|x_1|<d\}$, let $M=\sup_{\partial\Omega}|\psi|$ and $F=\sup_\Omega|f|$. The nonnegative barrier $v=M+(e^{2d}-e^{x_1+d})F$ satisfies $(\Delta+c)v\leq-F$. Apply the <weak maximum principle for elliptic operators> to $u-v$ and $-u-v$ to get $\|u\|_\infty\leq M+e^{2d}F$.