In Gilbarg-Trudinger's book 'Elliptic Partial Differential Equations of Second order', the maximum principle is used to derive the following gradient estimates for Poisson equations on Euclidean spaces. See Section 3.4; Eq. (3.15)

Let $u\in C^2(Q)\cap C^0(\bar Q)$ for the cube $Q=\{x \in \mathbb R^n \mid |x_i|<d\}$. Then we have the following gradient estimate: $$ |D_iu(0)| \le \frac{n}{d} \sup_{\partial Q} |u| + \frac{d}{2}\sup_Q |\Delta u| $$

However, in practice I am more interested in a possible analogous gradient estimate on some Riemannian manifold $(M,g)$.

Question: Let $\Omega\subset M$ be a bounded domain and $u\in C^2(\Omega)\cap C^0(\bar \Omega)$. Then can we find $c_1,c_2>0$ so that for $x\in M$ $$ \|\nabla^g u(x)\|_g \le c_1 \sup_{\partial \Omega} |u| + c_2 \mathrm{dist}(x,\partial \Omega) \cdot \sup_\Omega \|\Delta^g u\|_g $$

If this is true it should be some standard result. Do you know of any reference about this? Thanks!

*In general if I found some estimate in PDE book (e.g. Gilbarg-Trudinger), then when can we apply it to a Riemannian manifold?*