论文标题
周期图的非本地平均曲率流
The nonlocal mean curvature flow of periodic graphs
论文作者
论文摘要
我们建立了$ \ Mathbb {rmathbb {r}^n $的定期图中的非本地平均值曲率$ {α\ in（0,1）} $的良好性。此外，我们证明，如果解决方案最初在$ {\ rm h}^{1+β}（\ mathbb {t}^n）$中足够接近其积分平均值，则它在全球范围内存在并在及时迅速转化为常数。证据依赖于方程式的重新制定为准线性进化问题，该问题通过直接定位方法表明是抛物面类型的，以及对于此类问题的抽象抛物线理论。
We establish the well-posedness of the nonlocal mean curvature flow of order ${α\in(0,1)}$ for periodic graphs on $\mathbb{R}^n$ in all subcritical little Hölder spaces ${\rm h}^{1+β}(\mathbb{T}^n)$ with $β\in(0,1)$. Furthermore, we prove that if the solution is initially sufficiently close to its integral mean in ${\rm h}^{1+β}(\mathbb{T}^n)$, then it exists globally in time and converges exponentially fast towards a constant. The proofs rely on the reformulation of the equation as a quasilinear evolution problem, which is shown to be of parabolic type by a direct localization approach, and on abstract parabolic theories for such problems.
