Anisotropic mean curvature flow with contact angle and Neumann boundary conditions in arbitrary dimensions

TL;DR

The paper analyzes anisotropic mean curvature flow for graphs over a strictly convex domain with boundary conditions modeling prescribed contact angle or Neumann data. By exploiting degeneracy properties of the anisotropic operator $\sqrt{1+|Du|^2}\, D^2_{p_ip_j}F(Du,-1)$ and developing a robust gradient-estimate framework, it obtains time-uniform gradient bounds and long-time behavior results. Key steps include detailed degeneracy bounds for $D^2F$ and $D^3F$, a careful boundary/interior maximum principle analysis, and a translation-invariant asymptotic description via a related elliptic problem that yields a translating profile $w(x)+\lambda t$. The results extend known isotropic theories to the anisotropic setting, under smallness conditions on the anisotropy deviation from the isotropic case and on boundary data, with implications for the long-time convergence of anisotropic flows under boundary interactions. The work thus provides a rigorous path from a priori gradient control to convergence to translating solutions for both prescribed contact angle and Neumann problems in arbitrary dimensions.

Abstract

Over a bounded strictly convex domain in $\mathbb{R}^n$ with smooth boundary, we establish a priori gradient estimate for an anisotropic mean curvature flow with prescribed contact angle and Neumann boundary conditions. The estimates require careful analysis of the degeneracy property of the anisotropic mean curvature operator. As a result, for both problems, we can infer that the solutions converge to one that is translation invariant in time.

Figures (1)

• Figure 1: Left: $\theta<\pi/2$, liquid spreads out, Right: $\theta>\pi/2$, droplet.

Theorems & Definitions (20)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Theorem 3.1
• Lemma 3.2