Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain. Assume that $u(x)$ is the classical solution solving $$a_{ij}(x)\partial_{ij}u(x)+b_i(x)\partial_iu(x)+c(x)u(x)=f(x)$$ $$u(x)\Big|_{\partial \Omega}=g(x)$$ for some smooth enough coefficients and uniformly elliptic $a_{ij}$. I found that in Gilbarg and Trudinger's PDE book, Theorem 8.33 states that $$ |u|_{1,\alpha,\Omega'}\leq C(|u|_{0}+|g|_{0}+|f|_{0,\alpha}) $$ for $\Omega' \subset \Omega$ and some constant $C$ depending on the $C^{0,\alpha}$-norms of $a_{ij}$, $b_i$ and $c$ (and some other parameters).
Now, I am looking for a upper bound of $|\nabla u|$ on $\Omega'$, not the holder norm, can this upper be something like $$ \sup_{\Omega'}|u|\leq C(|u|_{0}+|g|_{0}+|f|_{0}) $$ for some constant $C$ depending on the $C^{0}$-norms of $a_{ij}$, $b_i$ and $c$, but not their holder norms? If this is to complicated, we can deal with the ODE case or equation in divergence form.