论文标题
薄的一相问题中的稳定锥体
Stable cones in the thin one-phase problem
论文作者
论文摘要
这项工作的目的是研究针对薄（或分数）一相不含边界问题的均匀稳定解决方案。 在尺寸上对稳定（或最小）均质解决方案进行分类的问题$ n \ geq3 $是完全开放的。在这种情况下，轴向对称的解决方案有望在最小表面的经典理论中扮演与西蒙斯锥的作用，但即使在这种简单的情况下，问题也是开放的。 本文的目标是双重的。一方面，我们的第一个主要贡献是首次找到薄单相问题的稳定性条件。令人惊讶的是，这需要将“大解决方案”用于分数laplacian，后者在自由边界上爆炸。 另一方面，使用我们的新稳定性条件，我们证明了任何轴向对称的均质稳定解决方案$ n \ le 5 $是一维的，独立于参数$ s \ in（0,1）$。
The aim of this work is to study homogeneous stable solutions to the thin (or fractional) one-phase free boundary problem. The problem of classifying stable (or minimal) homogeneous solutions in dimensions $n\geq3$ is completely open. In this context, axially symmetric solutions are expected to play the same role as Simons' cone in the classical theory of minimal surfaces, but even in this simpler case the problem is open. The goal of this paper is twofold. On the one hand, our first main contribution is to find, for the first time, the stability condition for the thin one-phase problem. Quite surprisingly, this requires the use of "large solutions" for the fractional Laplacian, which blow up on the free boundary. On the other hand, using our new stability condition, we show that any axially symmetric homogeneous stable solution in dimensions $n\le 5$ is one-dimensional, independently of the parameter $s\in (0,1)$.
