论文标题
亚历山德罗夫定理的一般非局部曲率：内核的几何影响
Alexandrov theorem for general nonlocal curvatures: the geometric impact of the kernel
论文作者
论文摘要
对于一般的径向对称，非增强的，非阴性的内核$ h \ in L ^ 1 _ {loc}（r ^ d）$，我们研究了$ r ^ d $中可测量集的刚性，并具有恒定的非局部$ h $ h $ -mean -mean-mean -mean-mean-mean-mean-mean-mean curvature。在$ h $的合适的“提高的可集成性”假设下，我们证明这些集合是有限的工会，只要满足自然的非修复条件，它们就会是相等的球。球的半径及其相互距离都可以从下面控制合适的参数，这明确取决于$ h $的水平集的度量。在最简单，常见的情况下，$ h $是正面的，有限的和减少的，我们的结果意味着任何有限的开放式设置或具有有限周长的任何有限的可测量集，具有恒定的非局部$ h $ - 均值曲率必须是一个球。
For a general radially symmetric, non-increasing, non-negative kernel $h\in L ^ 1 _{loc} ( R ^ d)$, we study the rigidity of measurable sets in $R ^ d$ with constant nonlocal $h$-mean curvature. Under a suitable "improved integrability" assumption on $h$, we prove that these sets are finite unions of equal balls, as soon as they satisfy a natural nondegeneracy condition. Both the radius of the balls and their mutual distance can be controlled from below in terms of suitable parameters depending explicitly on the measure of the level sets of $h$. In the simplest, common case, in which $h$ is positive, bounded and decreasing, our result implies that any bounded open set or any bounded measurable set with finite perimeter which has constant nonlocal $h$-mean curvature has to be a ball.
