 Solution

With the convention $Lu\ge 0$ for a classical subsolution, the weak maximum principle for elliptic operators isHere $L$ is uniformly elliptic with bounded drift, has no zeroth-order term, and the domain is bounded. Equivalently, $Lu\le 0$ gives ${\inf }_{\Omega }u\ge {\min }_{\partial \Omega }u$.

One can see why measurable coefficients suffice for the pointwise classical statement by adding $\epsilon e^{\kappa x_1}$. Choose $\kappa$ large enough that $\lambda \kappa >\Vert b_1{\Vert }_{\infty }$; then $L(e^{\kappa x_1})=e^{\kappa x_1}({\kappa }^2{a}_{11}+\kappa b_1)>0$. 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 $\epsilon \downarrow 0$ proves the stated estimate. Only the symmetric part of $a$ acts on the Hessian, and ellipticity gives its positive definiteness.