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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.05015 |
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| _version_ | 1866914241270775808 |
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| author | Lee, Sanghoon Lee, Taehun |
| author_facet | Lee, Sanghoon Lee, Taehun |
| contents | We establish a comparison principle for entire solutions of the Allen--Cahn equation whose nodal sets, possibly singular, are asymptotic to a regular minimizing hypercone. We show that inclusion of the positive phases enforces a global ordering of the solutions. As a consequence, the positive phase uniquely determines the solution, and strict phase inclusion implies that the corresponding nodal sets are disjoint. Our analysis relies on a maximum principle for the linearized operator on unbounded domains that are not necessarily smooth, and yields an Allen--Cahn analogue of the strong maximum principle for minimal hypersurfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05015 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nodal set comparison for Allen--Cahn solutions with conical asymptotics Lee, Sanghoon Lee, Taehun Analysis of PDEs Differential Geometry 35J61 (Primary) 35B08, 35B50, 53A10 (Secondary) We establish a comparison principle for entire solutions of the Allen--Cahn equation whose nodal sets, possibly singular, are asymptotic to a regular minimizing hypercone. We show that inclusion of the positive phases enforces a global ordering of the solutions. As a consequence, the positive phase uniquely determines the solution, and strict phase inclusion implies that the corresponding nodal sets are disjoint. Our analysis relies on a maximum principle for the linearized operator on unbounded domains that are not necessarily smooth, and yields an Allen--Cahn analogue of the strong maximum principle for minimal hypersurfaces. |
| title | Nodal set comparison for Allen--Cahn solutions with conical asymptotics |
| topic | Analysis of PDEs Differential Geometry 35J61 (Primary) 35B08, 35B50, 53A10 (Secondary) |
| url | https://arxiv.org/abs/2601.05015 |