In the paper by Guan Pengfei: "C^2 a priori estimates for degenerate Monge-Ampere equations" https://projecteuclid.org/euclid.dmj/1077242669 Prof. P. Guan proved in Theorem 1 that the degenerate Monge-Ampere equation satisfying Condition (C) on a strictly convex domain $\Omega$ with $\partial \Omega\in C^{2,1}$, with homogenous boundary value $\phi=0$ on $\partial \Omega$, has a unique $C^{1,1}$ convex solution.

And I now have some problem about the approximation argument used in the proof of Theorems 1 and 7.

In the statement of Theorem 1, one assumes the boundary $\partial \Omega \in C^{2,1}$. And Prof. P. Guan applied the approximation argument to show that the solvability of the Dirichlet problem holds under the weaker assumption on the regularity of the boundary $\partial \Omega \in C^{2,1}$. However, the a priori estimates (e.g. Lemma 10, Lemma 11 ect.) used by the author to prove Theorem 1 require the assumption on the boundary $\partial \Omega \in C^3$.

But I can not understand the approximate process in the proof of Theorem 1(see Pages 13 and 14) very well. Precisely, I do not understand why this approximate process works under the weaker assumption of $\partial \Omega\in C^{2,1}$. I believe this is well understood, but it still confuses me.

Thereby, I would like to ask for somebody to explain the detail why the existence works under the assumption of $\partial \Omega \in C^{2,1}$ rather than the assumption of $\partial \Omega \in C^3$ (as what the condition for the a priori estimates established in the same paper)?

Thanks!