论文标题
非本地平均曲率具有可集成内核的
Nonlocal Mean Curvature with Integrable Kernel
论文作者
论文摘要
我们研究了非本地曲率定义为$ h^j_Ω（x）：= \ int _ {\ mathbb {r}^n} j（x-y）（χ_{χ_{ω^c}（ω^c}（y） - χ_$ x $ x， $ω\ subset \ mathbb {r}^n $，$χ$是集合的特征函数，$ j $是径向对称，不合格的，非信息的卷积内核。几篇论文研究了具有不可整合奇点的非本地曲率的情况，这是对经典曲率概念的概括，该概念要求边界的规律性高于$ c^2 $。该形式的非局部曲率出现在许多不同的应用中，例如图像处理，曲率驱动运动，变形。在这项工作中，我们专注于通过集成核定义的恒定非本地曲率的问题。我们的结果为不可整合内核提供了一些扩展，使非局部框架中Alexandrov定理的恒定平均曲率问题是由两个单独的组独立建立的。 Ciraolo，Figalli，Maggi，Novaga和Cabré，Fall，Solà-Morales，Weth。使用非局部亚历山德罗夫定理，我们确定了针对不同的集成核的恒定平均非局部曲率的表面，作为位于距离$δ$分开的球，其中$δ$测量非局部相互作用的半径。
We study the prescribed constant mean curvature problem in the nonlocal setting where the nonlocal curvature has been defined as $$ H^J_Ω(x):=\int_{\mathbb{R} ^n} J(x-y)(χ_{Ω^c}(y)-χ_Ω(y))dy, $$ where $x \in \mathbb{R}^n$, $Ω\subset \mathbb{R}^n$, $χ$ is the characteristic function for a set, $J$ is a radially symmetric, nonegative, nonincreasing convolution kernel. Several papers have studied the case of nonlocal curvature with nonintegrable singularity, a generalization of the classical curvature concept, which requires the regularity of the boundary to be above $C^2$. Nonlocal curvature of this form appears in many different applications, such as image processing, curvature driven motion, deformations. In this work, we focus on the problem of constant nonlocal curvature defined via integrable kernel. Our results offer some extensions to the constant mean curvature problem for nonintegrable kernels, where counterparts to Alexandrov theorem in the nonlocal framework were established independently by two separate groups. Ciraolo, Figalli, Maggi, Novaga, and respectively, Cabré, Fall, Solà-Morales, Weth. Using the nonlocal Alexandrov's theorem we identify surfaces of constant mean nonlocal curvature for different integrable kernels as unions of balls situated at distance $δ$ apart, where $δ$ measures the radius of nonlocal interactions.
