Rigidity results for a triple junction solution of Allen-Cahn system
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915047744208896 |
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| author | Geng, Zhiyuan |
| author_facet | Geng, Zhiyuan |
| contents | For the two dimensional Allen-Cahn system with a triple-well potential, previous results established the existence of a minimizing solution $u:\mathbb{R}^2\rightarrow\mathbb{R}^2$ with a triple junction structure at infinity. We show that along each of three sharp interfaces, $u$ is asymptotically invariant in the direction of the interface and can be well-approximated by the 1D heteroclinic connections between two phases. Consequently, the diffuse interface is located in an $O(1)$ neighborhood of the sharp interface, and becomes nearly flat at infinity. This generalizes all the results for the triple junction solution with symmetry hypotheses to the non-symmetric case. The proof relies on refined sharp energy lower and upper bounds, alongside a precise estimate of the diffuse interface location. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02954 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rigidity results for a triple junction solution of Allen-Cahn system Geng, Zhiyuan Analysis of PDEs 35J47, 35J50, 35B08 For the two dimensional Allen-Cahn system with a triple-well potential, previous results established the existence of a minimizing solution $u:\mathbb{R}^2\rightarrow\mathbb{R}^2$ with a triple junction structure at infinity. We show that along each of three sharp interfaces, $u$ is asymptotically invariant in the direction of the interface and can be well-approximated by the 1D heteroclinic connections between two phases. Consequently, the diffuse interface is located in an $O(1)$ neighborhood of the sharp interface, and becomes nearly flat at infinity. This generalizes all the results for the triple junction solution with symmetry hypotheses to the non-symmetric case. The proof relies on refined sharp energy lower and upper bounds, alongside a precise estimate of the diffuse interface location. |
| title | Rigidity results for a triple junction solution of Allen-Cahn system |
| topic | Analysis of PDEs 35J47, 35J50, 35B08 |
| url | https://arxiv.org/abs/2412.02954 |