Flat level sets of Allen-Cahn equation in half-space

We prove a half-space Bernstein theorem for Allen-Cahn equation. More precisely, we show that every solution $u$ of the Allen-Cahn equation in the half-space $\overline{{\mathbb{R}}_{+}^n}:=\{(x_1,x_2,\cdots ,x_n)\in {\mathbb{R}}^n:x_1\ge 0\}$ with $\mathrm{\mid }u\mathrm{\mid }\le 1$, boundary value given by the restriction of a one-dimensional solution on $\{x_1=0\}$ and monotone condition ${\mathrm{\partial }}_{x_n}u>0$ as well as limiting condition ${\lim }_{x_n\to \pm \mathrm{\infty }}u(x^{\mathrm{\prime }},x_n)=\pm 1$ must itself be one-dimensional, and the parallel flat level sets and $\{x_1=0\}$ intersect at the same fixed angle in $(0,\frac{\pi }{2}]$.