In http://ams.rice.edu/leavingmsn?url=https://doi.org/10.1524/anly.1996.16.1.101 Prof. Xu-Jia Wang established the boundary estimates for second derivatives of the solution to classical Dirichlet problem for Monge-Ampere equations, in which the author only assumes $\partial \Omega$ and the boundary data $\varphi$ are both $C^3$. However, the author also gave counterexamples to show that if $\partial\Omega$ or boundary data $\varphi$ is only $C^{2,1}$ smooth, the solution may fail to be $C^2$ smooth near the boundary.

From Wang's counterexamples mentioned above and the approximation lead me to asking a question:

What is the optimal regularity assumptions on the boundary and boundary data such that Evans-Krylov theorem works under these assumptions.

I don't know whether the examples means that the optimal regularity assumption on $\varphi$ and $\partial\Omega$ for deriving $C^{2,\alpha}$ estimate up to boundary via Evans-Krylov theorem are both $C^3$?.

Thanks very much for your attention!