Consider two domains 
$$
\begin{aligned}
D_1&=\{x=(x_1,x_2,...,x_n)\in\mathbb{R}^n:x_n\leq 0\},\\
D_2&=\{x=(x_1,x_2,...,x_n)\in\mathbb{R}^n:x_n\leq \psi(x_1,x_2,...,x_{n-1})\},
\end{aligned}
$$
where $ \psi:\mathbb{R}^{n-1}\to\mathbb{R} $ is a smooth bounded function. I want to construct a conformal map $ \Phi:D_1\to D_2 $. I know that conformal maps between two arbitrary domains in high dimensional Euclidean space may not exist. Nevertheless here the domains are easy, I wonder if I can get such maps. If not, can I put more assumptions on it such that the result is true? Can you give me some hints or references?