论文标题
通过变异方法的周期性非局部CMC表面的存在和对称性
Existence and symmetry of periodic nonlocal-CMC surfaces via variational methods
论文作者
论文摘要
本文提供了周期非局部CMC表面存在的第一个变化证明。这些是经典Delaunay气缸的非局部类似物。更准确地说，我们显示了$ \ mathbb {r}^n $中的一个集合的存在，该集合在一个方向上是周期性的，在与该方向的平板正交中有一个处方（但任意的）体积，并具有恒定的非局部均值弯曲，并具有恒定的非局部均值曲率，并在体积约束下具有适当的分数元素的适当定期版本。此外，我们还表明，该集合是圆柱形对称的，更重要的是，它甚至在其一半时期也没有发光。这种单调性属性解决了一个空旷的问题和障碍物，这是其他作者早期尝试显示最小化物的存在的尝试。
This paper provides the first variational proof of the existence of periodic nonlocal-CMC surfaces. These are nonlocal analogues of the classical Delaunay cylinders. More precisely, we show the existence of a set in $\mathbb{R}^n$ which is periodic in one direction, has a prescribed (but arbitrary) volume within a slab orthogonal to that direction, has constant nonlocal mean curvature, and minimizes an appropriate periodic version of the fractional perimeter functional under the volume constraint. We show, in addition, that the set is cylindrically symmetric and, more significantly, that it is even as well as nonincreasing on half its period. This monotonicity property solves an open problem and an obstruction which arose in an earlier attempt, by other authors, to show the existence of minimizers.
