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PreprintarXiv licencearXiv

On the Allen--Cahn equation

  • Zhouyu Long
  • Wenming Zou
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Abstract

Using the framework developed by OpenAI, we prove that finite Morse index implies O(Rn−1) energy growth for bounded entire solutions of the quartic Allen--Cahn equation in dimensions 4≤ n≤9, uniformly in the center of the ball. Combined with the cited finite-index classification results, this gives planar rigidity for 4≤ n≤7 and, for nonconstant solutions, multiplicity-one stable minimal cone blowdowns through dimension nine. Within the framework, we derive an additive adjacent-layer interaction estimate, a weighted curvature and defect estimate with exponent p=9/4, and the localization needed for exterior stability. The limiting interaction coefficient $48$ is sharp in the unrestricted low-defect class with sufficiently large logarithmic selection depth.

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Cite

BibTeXRIS
BibTeX
@misc{long2026allencahn,
  title = {{On the Allen--Cahn equation}},
  author = {Zhouyu Long and Wenming Zou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28080},
  eprint = {2610.28080},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28080},
}
RIS
TY  - UNPB
TI  - On the Allen--Cahn equation
AU  - Zhouyu Long
AU  - Wenming Zou
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.28080
UR  - https://arxiv.org/abs/2610.28080
AB  - Using the framework developed by OpenAI, we prove that finite Morse index implies O(R^n−1) energy growth for bounded entire solutions of the quartic Allen--Cahn equation in dimensions 4≤ n≤9, uniformly in the center of the ball. Combined with the cited finite-index classification results, this gives planar rigidity for 4≤ n≤7 and, for nonconstant solutions, multiplicity-one stable minimal cone blowdowns through dimension nine. Within the framework, we derive an additive adjacent-layer interaction estimate, a weighted curvature and defect estimate with exponent p=9/4, and the localization needed for exterior stability. The limiting interaction coefficient $48$ is sharp in the unrestricted low-defect class with sufficiently large logarithmic selection depth.
ER  -