Alexandroff maximum principle (c.f. [21], Lemma 9.3)
Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain. Suppose $u\in C^{2}(\Omega)\cap C(\bar{\Omega})$. Denote $M=\sup_{\Omega}u-\sup_{\partial\Omega}u$. Define

(5.20)

$$\begin{split}\Gamma^{-}(u,\Omega)=\{&x\in\Omega:u(y)\leq u(x)+\nabla u(x)\cdot(y-x),\\ &\quad\quad\quad\quad\,\,\textrm{for any $y\in\Omega$ and }|\nabla u(x)|\leq\frac{M}{3diam\Omega}\}.\end{split}$$

Then for some dimensional constant $C_{d}>0$:

$$M\leq C_{d}\bigg(\int_{\Gamma^{-}(u,\Omega)}\det(-D^{2}u)dx\bigg)^{\frac{1}{d}}.$$

In particular, suppose $u$ satisfies $a_{ij}\partial_{ij}u\geq f$. Here $a_{ij}$ satisfies the ellipticity condition $a_{ij}\xi_{i}\xi_{j}\geq 0$. Define $D^{*}=(\det a_{ij})^{\frac{1}{d}}$. Then the following estimate holds:

(5.21)

$$M\leq C_{d}^{\prime}\,diam\,\Omega||\frac{f^{-}}{D^{*}}||_{L^{d}(\Omega)}.$$

Here $C_{d}^{\prime}$ is another dimensional constant.