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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 some expert 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!

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    $\begingroup$ The key point is that the estimates don't depend quantitatively on the $C^3$ regularity of the boundary, so you can approximate $\Omega$ with smoother $\Omega_k$ whose boundaries converge in the $C^{2,1}$ norm, and take the limit. You can probably do both approximations at once (letting $\varepsilon = 1/k$) or to keep it simpler, you could do the entire proof as written for $C^3$ domains, get a priori estimates, and finally do the domain approximation. $\endgroup$
    – user126920
    Commented Oct 8, 2018 at 15:07
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    $\begingroup$ The actual steps of the approximation procedure would be very similar to the one for linear elliptic equations, which you can find in Chapter 6 of Gilbarg-Trudinger. (I don't have the book in front of me, so I can't give a more precise reference.) $\endgroup$
    – user126920
    Commented Oct 8, 2018 at 15:20
  • $\begingroup$ @Stanley Snelson Thanks very much for your helpful comment! I believe that the approximation procedure you informed me is a good approach. $\endgroup$
    – xiaocha123
    Commented Oct 9, 2018 at 12:07
  • $\begingroup$ 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, $$ \det (D^2 u_\epsilon)=f_\epsilon \mbox{ in } \Omega, \mbox{ } u_\epsilon =0 \mbox{ on } \partial\Omega.$$ $\endgroup$
    – xiaocha123
    Commented Oct 9, 2018 at 12:45
  • $\begingroup$ 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! $\endgroup$
    – xiaocha123
    Commented Oct 9, 2018 at 12:55

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