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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 L1-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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Cite

BibTeXRIS
BibTeX
@misc{le2026isolation,
  title = {{Isolation of scalar Allen-Cahn local minimizers}},
  author = {Nam Q. Le},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11797},
  eprint = {2609.11797},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11797},
}
RIS
TY  - UNPB
TI  - Isolation of scalar Allen-Cahn local minimizers
AU  - Nam Q. Le
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.11797
UR  - https://arxiv.org/abs/2609.11797
AB  - 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.
ER  -