Convergence to the Grim Reaper for a Curvature Flow with Unbounded Boundary Slopes
Abstract
We consider a curvature flow $V=H$ in the band domain $\Omega :=[-1,1]\times \R$, where, for a graphic curve $\Gamma_t$, $V$ denotes its normal velocity and $H$ denotes its curvature. If $\Gamma_t$ contacts the two boundaries $\partial_\pm \Omega$ of $\Omega$ with constant slopes, in 1993, Altschular and Wu \cite{AW1} proved that $\Gamma_t$ converges to a {\it grim reaper} contacting $\partial_\pm \Omega$ with the same prescribed slopes. In this paper we consider the case where $\Gamma_t$ contacts $\partial_\pm \Omega$ with slopes equaling to $\pm 1$ times of its height. When the curve moves to infinity, the global gradient estimate is impossible due to the unbounded boundary slopes. We first consider a special symmetric curve and derive its uniform interior gradient estimates by using the zero number argument, and then use these estimates to present uniform interior gradient estimates for general non-symmetric curves, which lead to the convergence of the curve in $C^{2,1}_{loc} ((-1,1)\times \R)$ topology to the {\it grim reaper} with span $(-1,1)$.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2019
- DOI:
- 10.48550/arXiv.1907.11535
- arXiv:
- arXiv:1907.11535
- Bibcode:
- 2019arXiv190711535L
- Keywords:
-
- Mathematics - Differential Geometry;
- 35K93;
- 53C44;
- 35C07
- E-Print:
- Calculus of Variations and Partial Differential Equations (2021) 60: 159