We introduce several notions of generalised principal eigenvalue for a linear elliptic operator on a general unbounded domain, under boundary condition of the oblique derivative type. We employ these notions in the stability analysis of semilinear problems. Some of the properties we derive are new even in the Dirichlet or in the whole space cases. As an application, we show the validity of the hair-trigger effect for the Fisher-KPP equation on general, uniformly smooth domains.

中文翻译：

---

一般无界域中半线性抛物线问题的稳定性分析

我们在斜导数类型的边界条件下，在一般无界域上引入了线性椭圆算子的广义主特征值的几个概念。我们在半线性问题的稳定性分析中使用这些概念。我们推导出的一些性质即使在狄利克雷或整个空间情况下也是新的。作为一个应用，我们展示了 Fisher-KPP 方程在一般均匀光滑域上的触发效应的有效性。