Evans-Krylov theorem

Question

Do there exist estimates for nonconcave functionals similar to Evans-Krylov theorem in chapter 6 of Fully nonlinear elliptic equations by Luis A.C affarelli and Cabre? Perhaps there is a counterexample.

Answer 1

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 Gilbarg-Trudinger.

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 non-concave) but $u$ is not even $C^{1,1}$. (This implies that the $C^{1,\alpha}$ regularity that follows from Krylov-Safonov is the best we can hope for in general.) As far as I know, this question--whether 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 non-concave $F$. 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 Armstrong-Silvestre-Smart, 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 $C^1$.