= Solution
With the convention $Lu\geq0$ for a classical subsolution, the <weak maximum principle for elliptic operators> is
$$
\boxed{\sup_\Omega u\leq\max_{\partial\Omega}u,\qquad u\in C^2(\Omega)\cap C(\overline\Omega).}
$$
Here $L$ is uniformly elliptic with bounded drift, has no zeroth-order term, and the domain is bounded. Equivalently, $Lu\leq0$ gives $\inf_\Omega u\geq\min_{\partial\Omega}u$.
One can see why measurable coefficients suffice for the pointwise classical statement by adding $\varepsilon e^{\kappa x_1}$. Choose $\kappa$ large enough that $\lambda\kappa>\|b_1\|_\infty$; then $L(e^{\kappa x_1})=e^{\kappa x_1}(\kappa^2a_{11}+\kappa b_1)>0$. The perturbed subsolution cannot attain a maximum inside: its <gradient> would be zero and its Hessian <negative semidefinite>, forcing its elliptic second-order term to be nonpositive. Its maximum is therefore on the boundary. Letting $\varepsilon\downarrow0$ proves the stated estimate. Only the symmetric part of $a$ acts on the Hessian, and ellipticity gives its positive definiteness.
Back to article page
