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

TL;DR

This paper establishes Liouville-type theorems for capillary minimal graphs over half-spaces, showing under certain growth or boundedness conditions that such graphs must be flat, using gradient estimates and maximum principle adaptations.

Contribution

It provides new Liouville-type results for capillary minimal graphs with specific boundary conditions, extending classical flatness results to capillary settings.

Findings

Minimal graphs with linear growth are flat in specified dimensions and angles.

One-sided bounded minimal graphs over half-spaces are necessarily flat.

Gradient estimates for the mean curvature equation are crucial for the proofs.

Abstract

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 $n\geq4$ 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.