Half-space Liouville-type theorems for minimal graphs with capillary boundary

In this paper, we prove two Liouville-type theorems for capillary minimal graph over ${\mathbb{R}}_{+}^n$. First, if $u$ has linear growth, then for $n=2,3$ and for any $\theta \in (0,\pi )$, or $n\ge 4$ and $\theta \in (\frac{\pi }{6},\frac{5\pi }{6})$, $u$ must be flat. Second, if $u$ is one-sided bounded on ${\mathbb{R}}_{+}^n$, then for any $n$ and $\theta \in (0,\pi )$, $u$ must be flat. The proofs build upon gradient estimates for the mean curvature equation over ${\mathbb{R}}_{+}^n$ with capillary boundary condition, which are based on carefully adapting the maximum principle to the capillary setting.