In this paper we prove the uniqueness of the saddle-shaped solution $u\colon \mathbb{R}^{2m} \to \mathbb{R}$ to the semilinear nonlocal elliptic equation $(-\Delta)^\gamma u = f(u)$ in $\mathbb{R}^{2m}$, where $\gamma \in (0,1)$ and $f$ is of Allen–Cahn type. Moreover, we prove that this solution is stable if $2m\geq 14$. As a consequence of this result and the connection of the problem with nonlocal minimal surfaces, we show that the Simons cone $\{(x', x'') \in \mathbb{R}^{m}\times \mathbb{R}^m: |x'| = |x''|\}$ is a stable nonlocal $(2\gamma)$-minimal surface in dimensions $2m\geq 14$.

Saddle-shaped solutions of the fractional Allen–Cahn equation are doubly radial, odd with respect to the Simons cone, and vanish only in this set. It was known that these solutions exist in all even dimensions and are unstable in dimensions 2, 4, and 6. Thus, after our result, the stability remains an open problem only in dimensions 8, 10, and 12.

The importance of studying this type of solution is due to its relation with the fractional version of a conjecture by De Giorgi. Saddle-shaped solutions are the simplest non 1D candidates to be global minimizers in high dimensions, a property not yet established in any dimension.

中文翻译：

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分数阶Allen-Cahn方程鞍形解的唯一性和稳定性

在本文中，我们证明了鞍形解$ u \ colon \ mathbb {R} ^ {2m} \ to \ mathbb {R} $对于半线性非局部椭圆方程$（-\ Delta）^ \ gamma u的唯一性= $ \ mathbb {R} ^ {2m} $中的f（u）$，其中（\ 0,1）$和$ f $中的$ \ gamma是Allen–Cahn类型。此外，我们证明了如果$ 2m \ geq 14 $，则该解决方案是稳定的。由于该结果以及问题与非局部极小表面的联系，我们证明了西蒙斯锥$ \ {（x'，x''）\ in \ mathbb {R} ^ {m} \ times \ mathbb { R} ^ m：| x'| = | x''| \} $是稳定的非局部$（2 \ gamma）$最小曲面，尺寸为$ 2m \ geq 14 $。

分数Allen-Cahn方程的鞍形解是双径向的，相对于西蒙斯锥是奇数的，仅在该集合中才消失。众所周知，这些解决方案存在于所有偶数尺寸中，并且在尺寸2、4和6中不稳定。因此，根据我们的结果，稳定性仅在尺寸8、10和12中仍然是一个未解决的问题。

研究这类解决方案的重要性是由于它与De Giorgi猜想的分数形式有关。鞍形解决方案是在高维度上成为全局最小化器的最简单的非1D候选者，该属性尚未在任何维度上建立。