A height estimate for constant mean curvature graphs and uniqueness

TL;DR

This work addresses the height control of constant mean curvature (CMC) graphs on unbounded domains and its implications for uniqueness in the Dirichlet problem. The authors derive a height estimate using a moving sphere of radius $1/H$ and the maximum principle to bound the distance between boundary traces, yielding $d(F(\Gamma_1), F(\Gamma_2)) \le 2/H$ (with a weaker initial bound $\le 2a$). These height bounds translate into cross-section height bounds on domains $\Omega = \{(x,y): b_-(x) < y < b_+(x)\}$, enabling pointwise estimates of $u$ in terms of a fixed level with a universal constant $M'$ (e.g., $M' = 4M + 5/H$). Leveraging these estimates, the paper proves two uniqueness theorems for the Dirichlet problem on unbounded domains under growth conditions along sequences $x_n \to \infty$ (and, for the whole line, $x_n' \to -\infty$), via a Collin–Krust type bound for the difference of two solutions.

Abstract

In this paper, we give a height estimate for constant mean curvature graphs. Using this result we prove two results of uniqueness for the Dirichlet problem associated to the constant mean curvature equation on unbounded domains.

Figures (2)

• Figure 1:
• Figure 2:

Theorems & Definitions (18)

• Lemma 1
• proof
• Theorem 2
• Theorem 2
• proof
• proof : Proof of Theorem \ref{['estim']}
• Proposition 3
• proof
• Proposition 4
• proof