to a convex function $\tilde u$ defined in $\tilde\Omega$. We also have $\mathcal{G}_\varepsilon$ sub-converges in the Hausdorff distance to a convex surface $\mathcal{\tilde G}_0\in\mathbb{R}^3$.

\(iii) By a rotation of coordinates, the convex surface $\mathcal{\tilde G}_0$ satisfies $$\label{e405}
        \mathcal{\tilde G}_0\subset\{y_1\geq0\}\cap\{y_3\geq0\}$$ and $\mathcal{\tilde G}_0$ contains two segments $$\label{e406}
        \{(0,0,y_3)\,:\,0\leq y_3\leq 3\},\quad \{(y_1,0,0)\,:\,0\leq y_1\leq 1\}.$$

Hence, by (i)–(iii) we can assume that there is a sequence of solutions $\tilde u_k$ of $$\label{e407}
        U^{ij}w_{ij}=\varepsilon_k f\quad\mbox{in }\tilde\Omega_k,$$ where $w=[{\mbox{det}}\,D^2u]^{\alpha-1}$, and $\varepsilon_k\to0$ such that the normalized domain $\tilde\Omega_k$ converges to $\tilde\Omega$, $\tilde u_k$ converges to $\tilde u$ and the graph of $\tilde u_k$, denoted by $\mathcal{\tilde G}_k$ converges in the Hausdorff distance to $\mathcal{\tilde G}_0$.

Note that in $y$-coordinates, $\mathcal{\tilde G}_0$ is not a graph of a function near the origin. By adding some linear function to $\tilde u_k$ and $\tilde u$ and making a rotation of coordinates in $\mathbb{R}^3$, i.e., $z_i=R_{ij}y_j$, where $(R_{ij})$ is a $3\times3$ rotation matrix, $\mathcal{\tilde G}_k, \mathcal{\tilde G}_0$ can be represented by $z_3=v_k(z_1,z_2), z_3=v(z_1,z_2)$, respectively [@Z1]. Moreover, $v_k$ is a solution of the equation given in §2.4 near the origin, $v$ satisfies $$\label{e408}
        v\geq\frac12|z_1|,\quad\mbox{and}\quad v(z_1,0)=\frac12|z_1|.$$

As we know that $\mathcal{\tilde G}_k$ converges in the Hausdorff distance to $\mathcal{\tilde G}_0$, in the 