On stable solutions to the Allen-Cahn equation with bounded energy density in $\mathbb{R}^4$
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912568622186496 |
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| author | Florit-Simon, Enric Serra, Joaquim |
| author_facet | Florit-Simon, Enric Serra, Joaquim |
| contents | We show that stable solutions $u:\mathbb{R}^4\to (-1,1)$ to the Allen-Cahn equation with bounded energy density (or equivalently, with cubic energy growth) are one-dimensional.
This is known to entail important geometric consequences, such as robust curvature estimates for stable phase transitions, and the multiplicity one and Morse index conjectures of Marques-Neves for Allen-Cahn approximations of minimal hypersurfaces in closed 4-manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02739 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On stable solutions to the Allen-Cahn equation with bounded energy density in $\mathbb{R}^4$ Florit-Simon, Enric Serra, Joaquim Analysis of PDEs Differential Geometry 35J61, 35J20, 49Q05 We show that stable solutions $u:\mathbb{R}^4\to (-1,1)$ to the Allen-Cahn equation with bounded energy density (or equivalently, with cubic energy growth) are one-dimensional. This is known to entail important geometric consequences, such as robust curvature estimates for stable phase transitions, and the multiplicity one and Morse index conjectures of Marques-Neves for Allen-Cahn approximations of minimal hypersurfaces in closed 4-manifolds. |
| title | On stable solutions to the Allen-Cahn equation with bounded energy density in $\mathbb{R}^4$ |
| topic | Analysis of PDEs Differential Geometry 35J61, 35J20, 49Q05 |
| url | https://arxiv.org/abs/2509.02739 |