Some remarks on the Allen-Cahn equation in $\mathbb{R}^n$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910087947223040 |
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| author | Ferla, Gabriele |
| author_facet | Ferla, Gabriele |
| contents | In this short note we present new results on a higher-dimensional generalization of De~Giorgi's conjecture for Allen--Cahn type equations, focusing on dimensions $n \ge 9$. Although counterexamples are known in this regime, our goal is to identify assumptions on solutions that still enforce one-dimensional symmetry. We prove an analogue of Savin's theorem in arbitrary dimension: for energy-minimizing solutions whose level sets enjoy n-7 directions of monotonicity, we deduce one-dimensional symmetry. In the same spirit, we extend these ideas to nonlocal phase transitions, and we discuss an application to free boundary problems Finally, we establish a counterpart of the Ambrosio--Cabré theorem for solutions that are not necessarily energy minimizers and may lack bounded energy density, assuming instead n-2 directions of monotonicity everywhere. Overall, this note aims to further strengthen the connection between phase transition models and minimal surface theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_29060 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Some remarks on the Allen-Cahn equation in $\mathbb{R}^n$ Ferla, Gabriele Analysis of PDEs In this short note we present new results on a higher-dimensional generalization of De~Giorgi's conjecture for Allen--Cahn type equations, focusing on dimensions $n \ge 9$. Although counterexamples are known in this regime, our goal is to identify assumptions on solutions that still enforce one-dimensional symmetry. We prove an analogue of Savin's theorem in arbitrary dimension: for energy-minimizing solutions whose level sets enjoy n-7 directions of monotonicity, we deduce one-dimensional symmetry. In the same spirit, we extend these ideas to nonlocal phase transitions, and we discuss an application to free boundary problems Finally, we establish a counterpart of the Ambrosio--Cabré theorem for solutions that are not necessarily energy minimizers and may lack bounded energy density, assuming instead n-2 directions of monotonicity everywhere. Overall, this note aims to further strengthen the connection between phase transition models and minimal surface theory. |
| title | Some remarks on the Allen-Cahn equation in $\mathbb{R}^n$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.29060 |