Gradient estimate for elliptic equation

Question

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 $\frac{\Lambda}{\lambda}$?

Answer 1

Continuity of $d$ does not guarantee that $\nabla u$ is Holder. In fact, the regularity of $d$ is the best regularity one can expect for $\nabla u$.

To see this, consider the one-dimensional divergence-form equation $$(du')' = f.$$ Integrating and dividing by $d$, we see that $u'$ is at best the same regularity as $d$. (This is also an easy way to see why we have one less derivative in estimates for divergence-form equations than in those for nondivergence equations).

This also shows that the constant $C(d)$ depends on the modulus of continuity for $d$. Indeed, even if $d$ is $C^{\alpha}$ it can oscillate wildly and stay close to $1$, and in the model problem $u' = 1/d$ would have the same behavior.