论文标题
凸超曲面上的分数Michael-Simon Sobolev不平等
A fractional Michael-Simon Sobolev inequality on convex hypersurfaces
论文作者
论文摘要
经典的迈克尔·西蒙（Michael-Simon）和阿拉德（Allard）的不平等是在欧几里得空间的子手机上定义的功能的Sobolev不等式。它受歧管独立的通用常数管辖，但在右侧显示了额外的$ l^p $项，由基础歧管的平均曲率加权。我们在这里证明了这种不平等的分数版本，这是欧几里得空间的高度曲面，这是凸集的边界。它涉及该功能的gagliardo半确定性，以及其$ l^p $ norm norm fy hypersurface的分数平均曲率加权。 作为应用程序，我们在凸集的平滑分数平均曲率流中建立了一个新的上限，以实现最大生存时间。结合取决于初始集的周长，而不是其直径。
The classical Michael-Simon and Allard inequality is a Sobolev inequality for functions defined on a submanifold of Euclidean space. It is governed by a universal constant independent of the manifold, but displays on the right-hand side an additional $L^p$ term weighted by the mean curvature of the underlying manifold. We prove here a fractional version of this inequality on hypersurfaces of Euclidean space that are boundaries of convex sets. It involves the Gagliardo semi-norm of the function, as well as its $L^p$ norm weighted by the fractional mean curvature of the hypersurface. As an application, we establish a new upper bound for the maximal time of existence in the smooth fractional mean curvature flow of a convex set. The bound depends on the perimeter of the initial set instead of on its diameter.
