论文标题
在带有无界边界斜率的带状域中的曲率流上
On a Curvature Flow in a Band Domain with Unbounded Boundary Slopes
论文作者
论文摘要
We consider an anisotropic curvature flow $V= A(\mathbf{n})H + B(\mathbf{n})$ in a band domain $Ω:=[-1,1]\times R$, where $\mathbf{n}$, $V$ and $H$ denote the unit normal vector, normal velocity and curvature, respectively, of a graphic curve $γ_T$。当$ a> 0> b $和曲线$γ_t$ contacts $ \ partial_ \ pmω$，斜率等于$ \ pm 1 $ 1 $时（当解决方案移至无限时，斜率是无限的），我们会考虑这种情况。首先，我们介绍了全球范围的良好性，然后在$ a $ a和$ b $的一些对称假设下，我们显示了解决方案的统一内部梯度估计。基于这些估计值，我们证明$γ_t$将$ t \ to to \ to \ infty $收集到$ c^{2,1} _ {\ text {loc}}}（（（-1,1,1）\ times r）$ topology to {\ it Infinite}衍生物的cup like Wave in {\ it Infinite}衍生产品。
We consider an anisotropic curvature flow $V= A(\mathbf{n})H + B(\mathbf{n})$ in a band domain $Ω:=[-1,1]\times R$, where $\mathbf{n}$, $V$ and $H$ denote the unit normal vector, normal velocity and curvature, respectively, of a graphic curve $Γ_t$. We consider the case when $A>0>B$ and the curve $Γ_t$ contacts $\partial_\pm Ω$ with slopes equaling to $\pm 1$ times of its height (which are unbounded when the solution moves to infinity). First, we present the global well-posedness and then, under some symmetric assumptions on $A$ and $B$, we show the uniform interior gradient estimates for the solution. Based on these estimates, we prove that $Γ_t$ converges as $t\to \infty$ in $C^{2,1}_{\text{loc}} ((-1,1)\times R)$ topology to a cup-like traveling wave with {\it infinite} derivatives on the boundaries.
