Allen-Cahn Solutions with Triple Junction Structure at Infinity
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909182357143552 |
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| author | Sandier, Étienne Sternberg, Peter |
| author_facet | Sandier, Étienne Sternberg, Peter |
| contents | We construct an entire solution $U:\mathbb{R}^2\to\mathbb{R}^2$ to the elliptic system \[ ΔU=\nabla_uW(U), \] where $W:\mathbb{R}^2\to [0,\infty)$ is a `triple-well' potential. This solution is a local minimizer of the associated energy \[ \int \frac{1}{2}|\nabla U|^2+W(U)\,dx \] in the sense that $U$ minimizes the energy on any compact set among competitors agreeing with $U$ outside that set. Furthermore, we show that along subsequences, the `blowdowns' of $U$ given by $U_R(x):=U(Rx)$ approach a minimal triple junction as $R\to\infty$. Previous results had assumed various levels of symmetry for the potential and had not established local minimality, but here we make no such symmetry assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_13474 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Allen-Cahn Solutions with Triple Junction Structure at Infinity Sandier, Étienne Sternberg, Peter Analysis of PDEs We construct an entire solution $U:\mathbb{R}^2\to\mathbb{R}^2$ to the elliptic system \[ ΔU=\nabla_uW(U), \] where $W:\mathbb{R}^2\to [0,\infty)$ is a `triple-well' potential. This solution is a local minimizer of the associated energy \[ \int \frac{1}{2}|\nabla U|^2+W(U)\,dx \] in the sense that $U$ minimizes the energy on any compact set among competitors agreeing with $U$ outside that set. Furthermore, we show that along subsequences, the `blowdowns' of $U$ given by $U_R(x):=U(Rx)$ approach a minimal triple junction as $R\to\infty$. Previous results had assumed various levels of symmetry for the potential and had not established local minimality, but here we make no such symmetry assumptions. |
| title | Allen-Cahn Solutions with Triple Junction Structure at Infinity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2305.13474 |