We investigate stability properties of a type of periodic solutions of the N-vortex problem on general domains Ω ⊂ ℝ². The solutions in question bifurcate from rigidly rotating configurations of the whole-plane vortex system and a critical point a₀ ∈ Ω of the Robin function associated to the Dirichlet Laplacian of Ω. Under a linear stability condition on the initial rotating configuration, which can be verified for examples consisting of up to 4 vortices, we show that the linear stability of the induced solutions is solely determined by the type of the critical point a₀. If a₀ is a saddle, they are unstable. If a₀ is a nondegenerate maximum or minimum, they are stable in a certain linear sense. Since nondegenerate minima exist generically, our results apply to most domains Ω. The influence of the general domain Ω can be seen as a perturbation breaking the symmetries of the N-vortex system on ℝ². Symplectic reduction is not applicable and our analysis on linearized stability relies on the notion of approximate eigenvectors. Beyond linear stability, Herman’s last geometric theorem allows us to prove the existence of isoenergetically orbitally stable solutions in the case of N = 2 vortices.

中文翻译：

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一般域中N-涡旋问题周期解的稳定性

我们调查一个类型的周期解的稳定性能ň上一般域Ω⊂ℝ-vortex问题²。从整个平面涡流系统的刚性旋转的配置和临界点有问题分叉的溶液一个₀相关联，以Ω的狄利克雷拉普拉斯罗宾功能的∈Ω。在初始旋转构型的线性稳定性条件下，可以证明该示例包括多达4个涡旋，我们证明了诱导解的线性稳定性仅由临界点a ₀的类型决定。如果一个₀是鞍，他们是不稳定的。如果为₀是一个不变性的最大值或最小值，它们在某种线性意义上是稳定的。由于非退化极小值通常存在，因此我们的结果适用于大多数域Ω。一般域Ω的影响可以被看作是一种扰动破坏的对称性Ñ -vortex系统上ℝ ²。辛约简是不适用的，我们对线性稳定性的分析依赖于近似特征向量的概念。除了线性稳定性，Herman的最后一个几何定理使我们能够证明在N = 2涡旋的情况下等能量轨道稳定解的存在。