Do there exist estimates for nonconcave functionals similar to EvansKrylov theorem in chapter 6 of Fully nonlinear elliptic equations by Luis A.C affarelli and Cabre? Perhaps there is a counterexample.
In two variables, there are $C^{2,\alpha}$ estimates for fully nonlinear elliptic equations without any convexity/concavity assumptions on the nonlinearity. See, for example, section 17.3 of GilbargTrudinger.
In dimensions five and higher, it's false in general. There are counterexamples such as the one constructed by Nadirashvili and Vladut, where $u$ is a viscosity solution of $F(D^2u) = 0$ (with $F$ smooth and nonconcave) but $u$ is not even $C^{1,1}$. (This implies that the $C^{1,\alpha}$ regularity that follows from KrylovSafonov is the best we can hope for in general.) As far as I know, this questionwhether viscosity solutions of fully nonlinear elliptic equations are always $C^2$is still open in dimensions three and four.
There are some interesting results that give additional hypotheses under which we can salvage $C^{2,\alpha}$ regularity even though $F$ is not concave or convex. Caffarelli and Cabre have such a result for $F(D^2u,x) = 0$ where $F(\cdot,x)$ is the minimum of a concave and a convex function for each $x$. Yuan has a $C^{2,\alpha}$ estimate for a particular nonconcave $F$. Savin has a perturbative result for $F(D^2u, Du, u, x)$ saying that $u$ is $C^{2,\alpha}$ if $F$ is smooth and uniformly elliptic in a neighborhood of $(0,0,0,x)$ and $u$ is small in $L^\infty$. We should also mention ArmstrongSilvestreSmart, who showed $u$ solving $F(D^2u) = 0$ is $C^{2,\alpha}$ outside a set of Hausdorff dimension $n\varepsilon$, at least if $F$ is $C^1$.

$\begingroup$ Thanks for your patient answer. I will read these papers and look forward to commuinicating more with you. $\endgroup$ Sep 18 '18 at 3:14