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 $θ\in(0,π)$, or $n\geq4$ and $θ\in(\fracπ6,\frac{5π}6)$, $u$ must be flat. Second, if $u$ is one-sided bounded on $\mathbb{R}^n_+$, then for any $n$ and $θ\in(0,π)$, $u$ 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.

💡 Research Summary

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In this paper the authors investigate entire minimal graphs over the Euclidean half‑space ℝⁿ₊ that satisfy a capillary (constant contact‑angle) boundary condition on the flat boundary {x₁=0}. The graph Σ of a smooth function u:ℝⁿ₊→ℝ is required to solve the minimal surface equation
 div(∇u/√{1+|∇u|²}) = 0 in ℝⁿ₊,
and to meet the boundary with a fixed angle θ∈(0,π), which in analytic form reads
 u₁ = –cosθ √{1+|∇u|²} on ∂ℝⁿ₊.
The main goal is to prove two Liouville‑type theorems, i.e. to show that under mild growth or boundedness assumptions the only such graphs are affine (flat).

Theorem 1.1 (Liouville‑type I).
Assume u has linear growth, |u(x)| ≤ C₀(1+|x|).

1. If the dimension n equals 2 or 3, the theorem holds for every contact angle θ∈(0,π).
2. If n≥4, the result remains true provided θ belongs to the explicit interval
    U(n)= {θ∈(0,π) : |cosθ|² < (3n−7)(n−1) /