I am struggling with a problem like this: In dimension $n\geq 3$,
consider the following uniformly elliptic equation 


$$a^{ij}(x)u\_{ij}(x)+u\_{nn}=0$$


where $i,j = 1, \dots, n-1$, and $\lambda \operatorname{id} < (a^{ij}) < \Lambda \operatorname{id}$. Are there any interior gradient estimates? More specifically, I need the estimate $|u\_n|\leq C$ in the interior, where $C$ depends only on $|u|\_{L^\infty}$ and the elliptic constant. 



Interior gradient estimate for uniformly elliptic equations
For the constant of ellipticity $\nu=\lambda/\Lambda$ close to $1$
for elliptic equations of the form 
$$\sum\_{i,j=1}^na^{ij}(x)u\_{ij}(x)=0$$ Cordes proved in



> 
> Cordes, Heinz Otto. *Über die erste Randwertaufgabe bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen*. (German) Math. Ann. 131 (1956), 278—312. [MR0091400](http://www.ams.org/mathscinet-getitem?mr=0091400)
> 
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> 


that provided a solution is smooth enough it belongs locally to $C^{1+\beta\_0}$ where $\beta\_0\in(0,1)$ depends on $\nu$ and the estimate 
$$
\|\partial\_{x\_i} u\|\_{C^{\beta\_0}(B\_{1/2})}\le C(n,\nu)\sup\_{B\_1}|u|
$$
holds. So at least for smooth solutions the required estimate is true.


From the other hand for general $\nu\in(0,1)$, Safonov proved in



> 
> Safonov, M. V. *Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients*. (Russian) Mat. Sb. (N.S.) 132(174) (1987), no. 2, 275--288; [translation](http://www.turpion.org/php/paper.phtml?journal_id=sm&paper_id=3167) in Math. USSR-Sb. 60 (1988), no. 1, 269—281 [MR0882838](http://www.ams.org/mathscinet-getitem?mr=882838)
> 
> 
> 


for $n=3$ that such an estimate is not true. Namely there exists $\nu\_0\in(0,1)$ s.t. the estimate above does not holds for any $\nu\in(0,\nu\_0]$, $0<\beta<1$, $C>0$. Though the question posed is not directly answered by this result of Safonov it makes look like a a possibility that the answer can be negative at least for $\nu<\nu\_0$ and $n\ge4$ (where one can consider solutions not depending on $x\_n$.)


