Two dimensional anisotropic mean curvature flow with contact angle condition
Can Cui, Nung Kwan Yip
TL;DR
The paper analyzes anisotropic mean curvature flow for graphs over a strictly convex planar domain with a contact-angle boundary condition and extends the framework to Dirichlet problems in higher dimensions. It develops a priori gradient estimates to guarantee uniform ellipticity, enabling a well-posed parabolic theory for $u_t=G(Du,-1)D^2_{p_ip_j}F(Du,-1)u_{x_ix_j}$ with $D_N u=-\sqrt{1+|Du|^2}\cos\theta$, and constructs translating solutions via the elliptic problem $\lambda=G(Dw,-1)D^2_{p_ip_j}F(Dw,-1)w_{x_ix_j}$. The main contributions are a sharp gradient bound in $\mathbb{R}^2$ under near-isotropy and convexity assumptions, the existence/uniqueness of translating profiles, and the long-time convergence of solutions to these translating states (up to a constant). The Dirichlet problem in arbitrary $n$ is treated with corresponding gradient bounds and convergence results, highlighting the method's applicability beyond the planar case and clarifying the limitations in higher dimensions for the pure contact-angle approach. Overall, the work provides rigorous PDE-analytic control and asymptotics for anisotropic interface evolution under boundary constraints, with relevance to materials science and geometric flow theory.
Abstract
In this paper, we study surfaces which evolve by anisotropic mean curvature flow with contact angle boundary condition over a strictly convex domain in $\mathbb{R}^2$. We establish a prior gradient estimate for smooth solutions to this boundary value problem. The same approach can also handle Dirichlet boundary condition in $\mathbb{R}^n$, $n\geq 2$. For both problems, we prove that the solutions converge to one that is translation invariant in time.