= Solution
Assume uniform ellipticity $a^{ij}\xi_i\xi_j\geq\lambda|\xi|^2$ with $\lambda>0$, bounded coefficients, and $c\leq0$. The <weak maximum principle for elliptic operators> is
$$
\boxed{\sup_\Omega u\leq\max\{0,\sup_{\partial\Omega}u\}.}
$$
When $c=0$, zero can be omitted. The sign restriction is necessary, as shown by the <failure of the weak maximum principle with a positive zeroth-order coefficient>.
Let $M=\max(0,\sup_{\partial\Omega}u)$ and $w=u-M$, so $Lw=Lu-cM\geq0$ and $w\leq0$ on the boundary. For sufficiently large $k$, $q=e^{kx_1}$ satisfies
$$
Lq=q(k^2a^{11}+kb^1+c)>0.
$$
If $w$ is positive somewhere, $w+\varepsilon q$ retains a positive interior maximum above all its boundary values for small $\varepsilon>0$. At that point its <gradient> is zero and its <Hessian matrix> is negative semidefinite; ellipticity and $c\leq0$ give $L(w+\varepsilon q)\leq0$, contradicting its strict positivity. This <exponential perturbation proof of the weak maximum principle with drift> also works with $M=\sup_{\partial\Omega}u$ of either sign when $c=0$.
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