$\partial\Omega$ [@STW], then the above proof can be simplified.

In fact, the approximation in Lemma \[l301\] applies on any subdomain $\Omega'\subset\Omega$. Instead of considering the boundary $\varphi$, one can consider $$\mathcal{S}_{\Omega'}[u_0]=\{v\in C(\overline\Omega')\,:\,v \mbox{ convex}, v|_{\partial\Omega'}=u_0|_{\partial\Omega'}, Dv(\Omega')\subset Du_0(\overline\Omega'), v\geq u_0\},$$ and then obtain a local approximation sequence.

Smooth approximation
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Let $u$ be the maximizer of . From the obstacle approximation, $u$ is also the maximizer of over the set $$\label{e308}
        \mathcal{S}[\varphi,\Omega]=\left\{v\in C(\overline\Omega)\,:\,v \mbox{ convex}, v|_{\partial\Omega}=\varphi|_{\partial\Omega}, Dv(\Omega)\subset D\varphi(\overline\Omega)\right\}.$$ In this subsection, we prove that $u$ can be approximated by smooth solutions of $$\label{e309}
   U^{ij}w_{ij}=f(x,u),$$ where $U^{ij}$ is the cofactor of $D^2u$ and $w=[{\mbox{det}}\,D^2u]^{\alpha-1}$. This approximation enables us to apply the a priori estimates in Section 2.

\[l302\] Let $u$ be the maximizer of . Suppose $\partial\Omega$ is Lipschitz continuous. Then there exists a sequence of smooth solutions to equation converging locally uniformly to the maximizer $u$.

To prove the approximation, first we recall the existence and regularity of solutions of the following second boundary value problem [@TW05]. Let $B=B_R(0)$ be a ball such that $\Omega\Subset B_{R-1}(0)$ and $\phi$ is a smooth, uniformly convex function in $B$ and $\phi=c^*$ is constant on $\partial B$. Let $$\label{e310}
   H(t)=(1-t^2)^{-2n}$$ be a nonnegative smooth function 