Half-space minimizing solutions of a two dimensional Allen-Cahn system
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915701094088704 |
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| author | Geng, Zhiyuan |
| author_facet | Geng, Zhiyuan |
| contents | This paper studies minimizing solutions to a two dimensional Allen-Cahn system on the upper half plane, subject to Dirichlet boundary conditions, \begin{equation*}
Δu-\nabla_u W(u)=0, \quad u: \mathbb{R}_+^2\to \mathbb{R}^2,\ u=u_0 \text{ on } \partial \mathbb{R}_+^2, \end{equation*} where $W: \mathbb{R}^2\to [0,\infty)$ is a multi-well potential. We give a complete classification of such half-space minimizing solutions in terms of their blow-down limits at infinity. In addition, we characterize the asymptotic behavior of solutions near the associated sharp interfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24610 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Half-space minimizing solutions of a two dimensional Allen-Cahn system Geng, Zhiyuan Analysis of PDEs 35J50, 35B40, 35B44, 82B26 This paper studies minimizing solutions to a two dimensional Allen-Cahn system on the upper half plane, subject to Dirichlet boundary conditions, \begin{equation*} Δu-\nabla_u W(u)=0, \quad u: \mathbb{R}_+^2\to \mathbb{R}^2,\ u=u_0 \text{ on } \partial \mathbb{R}_+^2, \end{equation*} where $W: \mathbb{R}^2\to [0,\infty)$ is a multi-well potential. We give a complete classification of such half-space minimizing solutions in terms of their blow-down limits at infinity. In addition, we characterize the asymptotic behavior of solutions near the associated sharp interfaces. |
| title | Half-space minimizing solutions of a two dimensional Allen-Cahn system |
| topic | Analysis of PDEs 35J50, 35B40, 35B44, 82B26 |
| url | https://arxiv.org/abs/2512.24610 |