论文标题
平均曲率流，具有高度圆柱体的规定接触角
A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder
论文作者
论文摘要
在本文中，我们考虑了平均曲率流$ v = h+a $在高维缸$ω\ times \ r $中，其中，$ a $是一个常数，$ω$是$ \ r^n $中的有界域，对于高度$ y = y = y = u（x，x，x，t）$ y = $ f $ v $ and $ ch $ and cul and cul and cul and cul vel vel vel vel vel v y = U（x，x，x，x，t）假设HyperSurface接触缸边界$ \ partialω\ times \ r $带有规定的角度$θ（x）$。在某些假设下，例如$ω$严格凸出，$ \ | \cosθ\ | _ {c^2} $很小，或者$ω$不一定是凸的，但是$ | a | $很大，我们得出了一些{\ us in {\ us simplion-In-In-In-In-In-In-In-In-In-In-In-in-In-In-in-in-pime interient} bections}，以解决最初的边界值问题。 Then, we present a trichotomy result as well as its criterion for the asymptotic behavior of the solutions, that is, when $I:= A|Ω|+\int_{\partial Ω} \cosθ(x) dσ>0$ (resp. $=0$, $<0$), the solution $u$ converges as $t\to \infty$ to a translating solution with positive speed (resp. stationary solution, a以负速度翻译解决方案）。
In this paper we consider a mean curvature flow $V=H+A$ in a high dimensional cylinder $Ω\times \R$, where, $A$ is a constant, $Ω$ is a bounded domain in $\R^n$, and, for a hypersurface $y=u(x,t)$ over $Ω$, $V$ and $H$ denote its normal velocity and mean curvature, respectively. Assume the hypersurface contacts the cylinder boundary $\partial Ω\times \R$ with prescribed angle $θ(x)$. Under certain assumptions such as $Ω$ is strictly convex and $\|\cosθ\|_{C^2}$ is small, or $Ω$ is not necessarily convex but $|A|$ is sufficiently large, we derive some {\it uniform-in-time gradient bounds} for the solutions to initial boundary value problems. Then, we present a trichotomy result as well as its criterion for the asymptotic behavior of the solutions, that is, when $I:= A|Ω|+\int_{\partial Ω} \cosθ(x) dσ>0$ (resp. $=0$, $<0$), the solution $u$ converges as $t\to \infty$ to a translating solution with positive speed (resp. stationary solution, a translating solution with negative speed).
