Energy convexity and uniformity of the $H$-surface flow in $\mathbb{R}^{3}$ with Dirichlet boundary condition
Year: 2026
Type: preprint
Abstract: We prove that the energy functional associated with surfaces of prescribed mean curvature in $\mathbb{R}^3$ exhibits a convexity property when restricted to maps from the unit $2$-disk having small Dirichlet energy and a fixed boundary value, provided that the $3$-form $H \cdot \text{vol}_{\mathbb{R}^3}$ has a primitive satisfying certain bounds. Under milder assumptions on $H:\mathbb{R}^3 \to \mathbb{R}$, we show that an analogous convexity estimate holds along the heat flow of the functional (the $H$-surface flow) when the initial Dirichlet energy is sufficiently small. This is done first for classical solutions, and then extended by approximation to weak solutions using a quantitative uniqueness result adapted from previous work on the harmonic map heat flow. As a consequence of the convexity estimate, we show that the $H$-surface flow with small-energy initial map of class $C^0 \cap W^{1, 2}$ on the unit $2$-disk converges uniformly to a unique limit at infinite time, which solves the corresponding stationary problem (the $H$-surface system) with the same boundary value.
Source: arXiv (Cornell University)
Authors: Cheng, Da Rong, Lin, Longzhi, Zhou, Xin
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214671200