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 $Ω:=[-1,1]\times \R$, where, for a graphic curve $Γ_t$, $V$ denotes its normal velocity and $H$ denotes its curvature. If $Γ_t$ contacts the two boundaries $\partial_\pm Ω$ of $Ω$ with constant slopes, in 1993, Altschular and Wu \cite{AW1} proved that $Γ_t$ converges to a {\it grim reaper} contacting $\partial_\pm Ω$ with the same prescribed slopes. In this paper we consider the case where $Γ_t$ contacts $\partial_\pm Ω$ 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 $(-1,1)$.

Theorems & Definitions (13)

• Theorem 1.1
• Definition 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 4.1
• Theorem 4.2
• Lemma 5.1
• proof