The Allen-Cahn equation and general minimal cones

@misc{rico2026allencahn,
  title = {{The Allen-Cahn equation and general minimal cones}},
  author = {Oscar Ivan Agudelo Rico and Matteo Rizzi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.01335},
  eprint = {2610.01335},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.01335},
  note = {CC BY-NC-ND 4.0},
}

TY  - UNPB
TI  - The Allen-Cahn equation and general minimal cones
AU  - Oscar Ivan Agudelo Rico
AU  - Matteo Rizzi
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.01335
UR  - https://arxiv.org/abs/2610.01335
AB  - We construct solutions to the Allen-Cahn equation Δu+u−u^3=0 in ℝ^N+1, with N≥ 3, whose zero level set is asymptotic (at infinity) to a given minimal cone. Our construction is quite general, since we only use a suitable nondegeneracy assumption; neither stability nor symmetry of the underlying cone is required. Moreover, we calculate the morse index of such solutions. In particular, our main result generalizes a result by Pacard and Wei (2013) dealing with stable solutions to the Allen-Cahn equation in high dimensions. More precisely, we give a positive answer to a remark raised in the paper by Pacard and Wei about the possibility to relax the assumption for the underlying cone is area-minimizing or at least stable. As a corollary, we also obtain solutions with infinite morse index and connected zero level set.
ER  -