I am curious about the following question:

suppose $u$ is a solution to the uniformly elliptic equation $\sum_{i,j=1}^{n}a_{ij}(x)u_{ij}=f(x)$ in $\Omega$ and $u=0$ on$\partial \Omega$, where $\Omega$ is a bounded convex domain and for simplicity it is close to a unit ball in hausdorff distance, $a_{i,j}$ and $f(x)$ are smooth. $a_{ij}$ has largest eigenvalue $\alpha(x)=1$, and smallest eigenvalue $\beta(x)$.

is it possible to prove a $C^{2}$ estimate: $|D^{2}u|\leq C$ in the compact subdomain $\Omega'$ of $\Omega$, where $C$ depends on $|f|_{L^\infty}$ and the distance between $\partial \Omega$ and $\partial \Omega'$, but doesnt depend on the lower bound of $|\beta(x)|$?

The condition I forgot to put: Suppose $u$ is convex and smooth...

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