= Exponential perturbation proof of the weak maximum principle with drift
For $Lu=a^{ij}D_{ij}u+b^iD_iu$ with $a^{ij}\xi_i\xi_j\geq\lambda|\xi|^2$ and $|b^1|<\lambda\ell$, the function $q(x)=e^{-\ell x_1}$ satisfies
$$
Lq=q(\ell^2a^{11}-\ell b^1)>0.
$$
If $Lu\geq0$ had a strict interior maximum above the boundary maximum, then $u+\varepsilon q$ would retain an interior maximum for small $\varepsilon>0$, where ellipticity gives $L(u+\varepsilon q)\leq0$. This contradicts its strictly positive image under $L$.
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