Weak maximum principle for elliptic operators
= Weak maximum principle for elliptic operators
= Weak maximum principle
{synonym}
For $u\in C^2(\Omega)\cap C^0(\overline\Omega)$ and a uniformly elliptic second-order operator with bounded coefficients $Lu=a^{ij}D_{ij}u+b^iD_iu+cu$ with $c\leq0$, the inequality $Lu\geq0$ on a bounded domain implies
$$
\max_{\overline\Omega}u\leq\max\{0,\max_{\partial\Omega}u\}.
$$
An exponential barrier reduces the non-strict inequality to a contradiction at a positive interior maximum.