Minimizing solutions of degenerate Allen-Cahn equations with three wells in $\mathbb{R}^2$
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| author | Bronsard, Lia Sandier, Étienne Sternberg, Peter |
| author_facet | Bronsard, Lia Sandier, Étienne Sternberg, Peter |
| contents | We characterize all minimizers of the vector-valued Allen-Cahn equation in $\mathbb{R}^2$ under the assumption that the potential $W$ has three wells and that the associated degenerate metric does not satisfy the usual strict triangle inequality. These minimizers depend on one variable only in a suitable coordinate system.
In particular, we show that no minimizing solutions to $ Δu=\nabla W(u)$ on $\mathbb{R}^2$ can approach the three distinct values of the potential wells. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08111 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimizing solutions of degenerate Allen-Cahn equations with three wells in $\mathbb{R}^2$ Bronsard, Lia Sandier, Étienne Sternberg, Peter Analysis of PDEs 49, 35 We characterize all minimizers of the vector-valued Allen-Cahn equation in $\mathbb{R}^2$ under the assumption that the potential $W$ has three wells and that the associated degenerate metric does not satisfy the usual strict triangle inequality. These minimizers depend on one variable only in a suitable coordinate system. In particular, we show that no minimizing solutions to $ Δu=\nabla W(u)$ on $\mathbb{R}^2$ can approach the three distinct values of the potential wells. |
| title | Minimizing solutions of degenerate Allen-Cahn equations with three wells in $\mathbb{R}^2$ |
| topic | Analysis of PDEs 49, 35 |
| url | https://arxiv.org/abs/2509.08111 |