I was looking at Theorem 12.4 of Gilbarg and Trudinger's *Elliptic partial differential equations of second order* ([MR1814364](https://mathscinet.ams.org/mathscinet-getitem?mr=1814364), [Zbl 1042.35002](https://zbmath.org/1042.35002)):

> **Theorem 12.4**. Let $u$ be a bounded $C^2(\Omega)$ solution of $$ L u=a u_{x x}+2 b u_{x y}+c u_{y y}=f $$ where $L$ is uniformly elliptic,
> satisfying  $$\lambda\left(\xi^2+\eta^2\right) \leqslant a \xi^2+2 b \xi
 \eta+c \eta^2 \leqslant \Lambda\left(\xi^2+\eta^2\right) \quad
\quad \forall(\xi, \eta) \in \mathbb{R}^2 ;$$ where $\lambda$ and $\Lambda$
> denote the eigenvalues of the coefficient matrix and 
> $$\frac{\Lambda}{\lambda} \leqslant \gamma $$ in a domain $\Omega$ of
> $\mathbb{R}^2$. Then for some $\alpha=\alpha(\gamma)>0$, we have
> 
> $$ [u]_{1, \alpha}^* \leqslant C\left(|u|_0+|f/\lambda|_0^{(2)}\right), \quad C=C(\gamma).\label{1}\tag{12.22} $$

The authors remark at the end of this theorem:

> The significant feature of this result is that the estimate \eqref{1} [the gradient holder bound]
> depends only on bounds on the coefficients and not on any regularity
> properties. This is in contrast with the Schauder estimates (Theorem
> 6.2) which depend as well on the Hölder constants of the coefficients. The Hölder estimates of Chapter 8 for divergence from equations in $n$
> variables (Theorem 8.24) are also independent of the regularity
> properties of the coefficients, but those estimates concern the
> solution itself and not its derivatives. The validity of the analog of
> Theorem 12.4 for $n>2$ remains in doubt.


I was wondering if now there are known examples/counter-examples to this estimate in higher dimensions say $\Omega \subset \mathbb{R}^3?$