Comparing these work, I do not understand what is the difference between the two cases: the $C^{2,1}$ or $C^{3,1}$ regularity assumption. Namely, if the second derivatives blow up near the boundary point at which one of the $C^{4}$ conditions is not satisfied, then what happen and how does one get the $C^{3,\alpha}$ solutions? This puzzles me now. I appreciate the further information and explanation. Best regards

The boundedness of second estimates in the paper depends on the fourth order derivatives of $\partial\Omega$ and boundary data. As you pointed out, in Wang's examples, the second derivatives in fact blow up near a boundary point where one of the $C^3$ conditions is not satisfied (the 2-d Monge-Ampere equation in Wang's example is hence not uniformly elliptic). That is why the solution does not lie in $C^2$ near the boundary.

@John Wung Thanks for your helful and detailed answer. I still have a question on the related topic. As we know many authors studied the solvability of Monge-Ampere equation and general fully nonlinear equations. For instance, in the Theorem 1.2 of `On the Dirichlet problem for Hessian equations' projecteuclid.org/euclid.acta/1485890887 Prof. Turdinger proved the existence of $C^{3,\alpha}$ solutions for Dirichlet problem of equations on $\Omega\subset\mathbb{R}^n$, provided that $\partial\Omega$ and boundary data are both $C^{3,1}$, and other assumptions hold.

Thank you very much

However, in my opinion, the author did not mention definitely what are the smooth exhausting domains used by himself (maybe I made the mistake here). Hence, in my opinion, it seems that the approximating process in the paper is different from that is presented in Gilbarg-Trudinger's book (see Chapter 8, probably, Theroem 8.34). Precisely, the approximating Dirichlet problems (55) in the paper are still defined on $\Omega$, rather than on certain smooth and strictly convex exhausting domains of $\Omega$ itself. That is the part which confuses me. Thanks!

But there is still a problem confusing me. Firstly, I shall present my own understanding: the author verified that the approximating functions $f_{\epsilon,\rho}$ satisfy Condition (C) near the boundary $\partial \Omega$ such that one can obtain the approximating Dirichlet problems (55) by letting $\rho \rightarrow 0^+$, that is, $$ \det (D^2 u_\epsilon)=f_\epsilon \mbox{ in } \Omega, \mbox{ } u_\epsilon =0 \mbox{ on } \partial\Omega.$$

@Stanley Snelson Thanks very much for your helpful comment! I believe that the approximation procedure you informed me is a good approach.