发表机构
Indiana University（印第安纳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明带双阱势和齐次 Neumann 边界条件的 Allen-Cahn 泛函的标量局部极小值在 L1 拓扑中孤立，通过解析 Lyapunov-Schmidt 约化肯定回答了 Kohn 和 Sternberg 的问题。
AI 中文摘要
我们证明，在有界 Lipschitz 域上，具有双阱势和齐次 Neumann 边界条件的 Allen-Cahn 泛函的标量局部极小值在 $L^1$-拓扑中是孤立的。这肯定地回答了 Kohn 和 Sternberg 提出的一个问题（Local minimisers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989)）。证明实现了一种解析的 Lyapunov-Schmidt 约化，其灵感来自 Jiang-Yu 和 Hirsch-Smith 在椭圆环境中关于单调动力系统理论的工作。
英文摘要
We show that scalar local minimizers of the Allen-Cahn functional with the double-well potential and homogeneous Neumann boundary conditions on a bounded Lipschitz domain are isolated in the $L^1$-topology. This affirmatively answers a question raised by Kohn and Sternberg (Local minimisers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989)). The proof implements an analytic Lyapunov-Schmidt reduction inspired by the theory of monotone dynamical systems in works of Jiang-Yu and Hirsch-Smith in an elliptic setting.
Comments17 pages. The main results of this paper were achieved through a series of inquiries with OpenAI's GPT-6 Astra. The key strategies using the analytic Lyapunov-Schmidt reduction inspired by the works of Jiang-Yu and Hirsch-Smith were obtained by ChatGPT. The author checked, reworked, and simplified all arguments and rewrote the proofs of the main results. v2: minor edits
