A question about the $C^{2,\alpha}$ regularity of concave fully nonlinear uniformly elliptic equation

Question

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.

Answer 1

A linear uniformly elliptic equation of the form $F(D^2u) = 0$ can only be $\Delta u = 0$ (up to an affine transformation of the solution) since the only inputs are the second derivatives. Equations like $a^{ij}(x)u_{ij} = 0$ are of the form $F(D^2u,x) = 0$. If $F(.,x_0)$ is concave for each $x_0$ (corresponding to freezing the coefficient in the linear setting) and $F$ is Holder continuous in $x$, then we get $C^{2,\alpha}$ regularity by applying Evans-Krylov and a perturbation argument (see ch. 8 of Caffarelli-Cabre). This is a generalization of Schauder estimates.

A more interesting concave equation "built out of" linear ones is the Bellman equation, $$\inf_{\alpha}a^{ij}_{\alpha}u_{ij} = 0$$ where $a^{ij}_{\alpha}$ is a collection of uniformly elliptic constant-coeffient matrices.

For the second question, the proof actually gives $C^{2,\alpha}$ regularity for a $C^{1,1}$ viscosity solution in all of $B_{1/2}$. Indeed, one shows the oscillation decay of $D^2u$ (as an $L^{\infty}$ map) $$diam(D^2u(B_{\delta^k}(x))) < 2^{-k}\|u\|_{C^{1,1}(B_1)}$$ for some universal $\delta$. It follows that $D^2u$ is Holder continuous up to modification on a set of measure $0$. By using quadratic approximations to $u$ at nearby points it is straightforward to show that $u$ is in fact $C^{2,\alpha}$ everywhere. (Think for example that the function $\chi_{\{0\}}$ cannot be the derivative of any differentiable function, since if it were it would lie close to a line of slope $1$ near $0$ and violate being locally constant away from 0).

As a side note, it is an interesting result of Armstrong, Silvestre and Smart that for any uniformly elliptic equation $F(D^2u) = 0$ with $F \in C^1$ we get $C^{2,\alpha}$ regularity off of a set of Hausdorff dimension $n-\epsilon$ for some small universal $\epsilon$.