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arXiv · 2609.11797

Isolation of scalar Allen-Cahn local minimizers

Nam Q. Le
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Abstract

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.

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Nam Q. Le. 2026-09-10. Isolation of scalar Allen-Cahn local minimizers. https://arxiv.org/abs/2609.11797

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