While reading Theorem 6.6 of Chapter Six of "Fully nonlinear elliptic equation" by Luis A. Caffarelli and Xavier Cabre in the American mathematical society colloquium publications vol. 43, I get two problems as follow.
The theorem 6.6 of this chapter is to prove the $C^{2,\alpha}$ regularity of the viscosity solution of the concave fully nonlinear uniformly elliptic equation $F(D^2u)=0$, where $F$ is just a concave function defined on the symmetric matrices and F is not required to be differentiable.
My first problem is that the uniform linear elliptic operator is special concave uniform elliptic, and in order to get $C^{2,\alpha}$ of the solution, we need the $C^{0,\alpha}$ differentiability of the coefficients of the operator. But here the theorem states that we can get the $C^{2,\alpha}$ regularity of the viscosity solution without any differentiability of the equation. I guess the reason may be that we consider the viscosity solution here, but I'm not sure whether we can get the $C^{2,\alpha}$ regularity of the classical solution of the concave uniformly elliptic equation. (This may be obvious wrong, but I want to know if the reason is just that we consider the viscosity solution.)
The second problem is the proof given by the book. In order to prove the theorem, the author applied the Evans-Krylov theorem to the $C^{1,1}$ viscosity solution of the equation $F(D^2 u)=0$, and the $C^{1,1}$ regularity of such an equation is obtained in the proof. While applying the process of proving the Evans-Krylov theorem to the $C^{1,1}$ viscosity solution, we can just get the $C^{2,\alpha}$ regularity in the subset $B_{\frac{1}{2}}\cap A$ of $B_{\frac{1}{2}}$, where $B_{\frac{1}{2}}-A$ is of measure zero. I wonder how can we get the regularity of the viscosity solution in $B_{\frac{1}{2}}$ without any other assumption. In the paper of Evans, the smoothness of $F$ is needed in order to use the method of continuity, and Evans considered the $C^{2,\alpha}$ regularity of the classic solution.
For the second problem, I have tried to use the smooth concave uniformly elliptic operator $F_k$ to approximate $F$, but in order to get a convergent viscosity solution of $F$ from the limit of $u_k$ that is the viscosity solution of $F_k$, we need a uniform bound for $u_k$, which is an obstruction for me now.
