Given:

1)a bounded domain $\Omega$ in $\mathbb R^n$ of class $\mathcal{C}^{\infty}$

2) the function $f\in L^{\infty}(\Omega)$ with $\int_{\Omega} f=0$

3)$g=(g_i,\ldots,g_n)\in \mathcal{C}^\alpha(\Omega)$

4) a scalar diffusion coefficient $d \in \mathcal{C} ^0(\Omega)$ s.t. $0<\lambda<d(x)<\Lambda<+\infty \quad \forall x\in \Omega$

there exists an estimate for the $\mathcal{C}^{1,\alpha}(\Omega)$-norm of $u$, weak solution of the elliptic equation

$\int_{\Omega}d\ \nabla u \cdot \nabla \Phi=\int_\Omega f \Phi +g \cdot \nabla \Phi \quad \forall \Phi \in H^1(\Omega)$

with Neumann boundary condition $d\nabla u \cdot n_{\partial \Omega}=0$ and with the condition $\int u=0$ (to fix the solution up to additive constant).

I found in theorem 8.33 and 8.34 in the book of Gilbarg and Trudinger "Elliptic Partial Differential Equations of Second Order" an estimate for the above equation but with Dirichlet boundary condition $u=\varphi\in \mathcal{C}^{1,\alpha}(\partial \Omega) \ on \ \partial \Omega$ of type

$\|u\|_{\mathcal{C}^{1,\alpha}(\Omega)}\leq C(d)(\|u\|_{C^0}+\|\varphi\|_{\mathcal{C}^{1,\alpha}}+\|f\|_{L^{\infty}}+\|g\|_{\mathcal{C}^{0,\alpha}})$

and assuming $d \in \mathcal{C}^{0,\alpha}$.

My questions now are:

1)"Is it possibile to have an estimate of type

$\|u\|_{\mathcal{C}^{1,\alpha}(\Omega)}\leq C(d)(\|f\|_{L\infty}+\|g\|_{\mathcal{C}^0,\alpha})$ having just $d\in \mathcal{C}^0(\Omega)$?

2) What can be say about the constant $C(d)$? Is it possible possible to say that is proportinal to $\frac{\Lambda}{\lambda}$?