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arXiv 2002.04549math.AP
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具有无界边界斜率的带状区域中的曲率流

On a Curvature Flow in a Band Domain with Unbounded Boundary Slopes

Lixia Yuan, Wei Zhao

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摘要：

考虑带域$Ω:=[-1,1]\times R$中的各向异性曲率流$V= A(\mathbf{n})H + B(\mathbf{n})$，其中$\mathbf{n}$，$V$和$H$分别表示曲线$Γ_t$的单位法向量，法速度和曲率.我们考虑$A>0>B$和曲线$Γ_t$与$\partial_\pm Ω$接触的情况，其斜率等于其高度的$\pm 1$倍（当解移动到无穷大时，其是无界的）。首先，我们提出了整体适定性，然后，在一些对称的假设下，$A$和$B$，我们显示了一致的内部梯度估计的解决方案。基于这些估计，我们证明了$Γ_t$收敛为$t\to \infty$在$C^{2,1}_{\text{loc}} ((-1,1)\times R)$拓扑的杯状行波与{ \it无限}衍生物的边界。

英文摘要：

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 $Γ_t$. 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.

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