strictly convex near $\partial\Omega$, $u$ can be recovered from $u^*$ by the same transform. If $u$ is $C^2$ smooth at $x$, $y=Du(x)$ and ${\mbox{det}}\,D^2u(x)\neq0$, then the Hessian matrix $D^2u(x)$ is the inverse of the Hessian matrix $D^2u^*(y)$, and $$\label{e208}
        {\mbox{det}}\,D^2u(x)=[{\mbox{det}}\,D^2u^*(y)]^{-1}.$$ In particular, if $u$ is a maximizer of the functional $J_\alpha$, $u^*$ is a maximizer of the dual functional $$\label{e209}
        J^*_\alpha(u)=
        \begin{cases}
\ \int_{\Omega^*}[{\mbox{det}}D^2u^*]^{1-\alpha}\, dy 
-\alpha\int_{\Omega^*}f(Du^*)(yDu^*-u^*){\mbox{det}}\,D^2u^*\,dy, \ \ \alpha>0 \ 
\text{and}\ \alpha\neq 1,&\\[5pt]
\ -\int_{\Omega^*}{\mbox{det}}D^2u^*\log{\mbox{det}}D^2u^*\,dy
-\int_{\Omega^*}f(Du^*)(yDu^*-u^*){\mbox{det}}\,D^2u^*\,dy,\ \ \alpha =0.&
\end{cases}$$ Therefore, if $u^*$ is smooth, it satisfies the equation $$\label{e210}
        U^{*ij}w^*_{ij}=
        \begin{cases}
        -\frac{\alpha}{1-\alpha}f(Du^*){\mbox{det}}D^2u^*, &\ \alpha>0 \
\text{and}\ \alpha\neq 1,\\[5pt]
   -f(Du^*){\mbox{det}}D^2u^*, &\ \alpha=0,
        \end{cases}$$ where $U^{*ij}$ is the cofactor matrix of $D^2u^*$ and $$\label{e211}
w^*=
\begin{cases}
\ [{\mbox{det}}D^2u^*]^{-\alpha}, & \alpha>0 \
\text{and}\ \alpha\neq 1,\\[5pt]
\ -\log{\mbox{det}}D^2u^*,& \alpha =0.
\end{cases}$$

By a similar argument to that of Lemma \[l202\], we have the following result [@TW00; @TW05; @Z1].

\[l203\] Let $u^*$ be a smooth convex solution of in $\Omega^*$ in dimension 2, $u^*=0$ on $\partial\Omega^*$. Then for any $y\in\Omega^*$, we have the a priori estimate $$\label{e212}
        {\mbox{det}}\,D^2u^*(y)\leq C,$$ where $C$ depends only on $\alpha, {\mbox{dist}}(y,\partial\Omega^*), \sup_{\Omega^*}|u^*|, \sup_{\Omega^*}|Du^*|$, and $\inf f$.

By and , we have ${\mbox{det}}\,D^2u\geq C$ has a positive lower bound. Note 